In mathematics , the L spaces are function spaces defined using a natural generalization of the p -norm for finite-dimensional vector spaces . They are sometimes called Lebesgue spaces , named after Henri Lebesgue ( Dunford & Schwartz 1958 , III.3), although according to the Bourbaki group ( Bourbaki 1987 ) they were first introduced by Frigyes Riesz ( Riesz 1910 ).
231-443: L spaces form an important class of Banach spaces in functional analysis , and of topological vector spaces . Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. In statistics, measures of central tendency and statistical dispersion , such as
462-424: A ȷ ^ | u ( i y ) u ( θ ) | = | exp ( log | a ȷ ^ | p θ p 0 ) exp ( − i y log |
693-1480: A ȷ ^ | p θ ( 1 p 0 − 1 p 1 ) ) | = | a ȷ ^ | p θ p 0 = | f ( ξ ) | p θ p 0 {\displaystyle {\begin{aligned}\left\vert f_{\mathrm {i} y}(\xi )\right\vert &=\left\vert \left\vert a_{\hat {\jmath }}\right\vert ^{\frac {u(\mathrm {i} y)}{u(\theta )}}\right\vert \\&=\left\vert \exp {\biggl (}\log \left\vert a_{\hat {\jmath }}\right\vert {\frac {p_{\theta }}{p_{0}}}{\biggr )}\exp {\biggl (}-\mathrm {i} y\log \left\vert a_{\hat {\jmath }}\right\vert p_{\theta }{\biggl (}{\frac {1}{p_{0}}}-{\frac {1}{p_{1}}}{\biggr )}{\biggr )}\right\vert \\&=\left\vert a_{\hat {\jmath }}\right\vert ^{\frac {p_{\theta }}{p_{0}}}\\&=\left\vert f(\xi )\right\vert ^{\frac {p_{\theta }}{p_{0}}}\end{aligned}}} which implies that ‖ f i y ‖ p 0 ≤ ‖ f ‖ p θ p θ p 0 {\textstyle \lVert f_{\mathrm {i} y}\rVert _{p_{0}}\leq \lVert f\rVert _{p_{\theta }}^{\frac {p_{\theta }}{p_{0}}}} . With
924-524: A , b ∈ A . {\displaystyle a,b\in A.} If X {\displaystyle X} is a normed space and K {\displaystyle \mathbb {K} } the underlying field (either the real or the complex numbers ), the continuous dual space is the space of continuous linear maps from X {\displaystyle X} into K , {\displaystyle \mathbb {K} ,} or continuous linear functionals . The notation for
1155-453: A , b ) ↦ a b ∈ A {\displaystyle A\times A\ni (a,b)\mapsto ab\in A} is continuous. An equivalent norm on A {\displaystyle A} can be found so that ‖ a b ‖ ≤ ‖ a ‖ ‖ b ‖ {\displaystyle \|ab\|\leq \|a\|\|b\|} for all
1386-512: A canonical factorization of T {\displaystyle T} as T = T 1 ∘ π , T : X ⟶ π X / ker ( T ) ⟶ T 1 Y {\displaystyle T=T_{1}\circ \pi ,\ \ \ T:X\ {\overset {\pi }{\longrightarrow }}\ X/\ker(T)\ {\overset {T_{1}}{\longrightarrow }}\ Y} where
1617-422: A linear operator that boundedly maps L ( μ 1 ) into L ( μ 2 ) and L ( μ 1 ) into L ( μ 2 ) . For 0 < θ < 1 , let p θ , q θ be defined as above. Then T boundedly maps L ( μ 1 ) into L ( μ 2 ) and satisfies the operator norm estimate In other words, if T is simultaneously of type ( p 0 , q 0 ) and of type ( p 1 , q 1 ) , then T
1848-412: A measure space by giving it the discrete σ-algebra and the counting measure . Then the space ℓ p ( I ) {\displaystyle \ell ^{p}(I)} is just a special case of the more general L p {\displaystyle L^{p}} -space (defined below). An L p {\displaystyle L^{p}} space may be defined as
2079-411: A metric . The metric space ( R n , d p ) {\displaystyle (\mathbb {R} ^{n},d_{p})} is denoted by ℓ n p . {\displaystyle \ell _{n}^{p}.} Although the p {\displaystyle p} -unit ball B n p {\displaystyle B_{n}^{p}} around
2310-1394: A metric space ( X , d ) . {\displaystyle \ (X,d)~.} A sequence x 1 , x 2 , … {\displaystyle \ x_{1},x_{2},\ldots \ } is called Cauchy in ( X , d ) {\displaystyle \ (X,d)\ } or d {\displaystyle d} -Cauchy or ‖ ⋅ ‖ {\displaystyle \ \|\cdot \|} -Cauchy if for every real r > 0 , {\displaystyle \ r>0\ ,} there exists some index N {\displaystyle N} such that d ( x n , x m ) = ‖ x n − x m ‖ < r {\displaystyle \ d\left(x_{n},x_{m}\right)=\left\|x_{n}-x_{m}\right\|<r\ } whenever m {\displaystyle m} and n {\displaystyle n} are greater than N . {\displaystyle \ N~.} The normed space ( X , ‖ ⋅ ‖ ) {\displaystyle \ (X,\|\cdot \|)\ }
2541-474: A norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} that induces on X {\displaystyle X} the topology τ {\displaystyle \tau } and also makes ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} into a Banach space. A Hausdorff locally convex topological vector space X {\displaystyle X}
SECTION 10
#17327866355152772-859: A simple function , that is f = ∑ j = 1 m a j 1 A j {\displaystyle f=\sum _{j=1}^{m}a_{j}\mathbf {1} _{A_{j}}} for some finite m ∈ N {\textstyle m\in \mathbb {N} } , a j = | a j | e i α j ∈ C {\textstyle a_{j}=\left\vert a_{j}\right\vert \mathrm {e} ^{\mathrm {i} \alpha _{j}}\in \mathbb {C} } and A j ∈ Σ 1 {\textstyle A_{j}\in \Sigma _{1}} , j = 1 , 2 , … , m {\textstyle j=1,2,\dots ,m} . Similarly, let g {\textstyle g} denote
3003-575: A translation invariant distance function , called the canonical or ( norm ) induced metric , defined for all vectors x , y ∈ X {\displaystyle \ x,y\in X\ } by d ( x , y ) := ‖ y − x ‖ = ‖ x − y ‖ . {\displaystyle \ d(x,y):=\|y-x\|=\|x-y\|~.} This makes X {\displaystyle X} into
3234-724: A vector space when addition and scalar multiplication are defined pointwise. That the sum of two p {\displaystyle p} -th power integrable functions f {\displaystyle f} and g {\displaystyle g} is again p {\displaystyle p} -th power integrable follows from ‖ f + g ‖ p p ≤ 2 p − 1 ( ‖ f ‖ p p + ‖ g ‖ p p ) , {\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),} although it
3465-423: A (non-complete) pre-Hilbert vector subspace of ℓ 2 ( N ) . {\displaystyle \ell ^{2}(\mathbb {N} ).} This norm-induced topology also makes ( X , τ d ) {\displaystyle \ \left(X,\tau _{d}\right)\ } into what is known as a topological vector space (TVS), which by definition
3696-443: A Banach space X {\displaystyle X} is canonically a metric Banach manifold modeled on X {\displaystyle \ X\ } since the inclusion map U → X {\displaystyle \ U\to X\ } is an open local homeomorphism . Using Hilbert space microbundles , David Henderson showed in 1969 that every metric manifold modeled on
3927-463: A Banach space, the space B ( X , Y ) {\displaystyle B(X,Y)} is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps ; in that case the space B ( X , Y ) {\displaystyle B(X,Y)} reappears as a natural bifunctor . If X {\displaystyle X}
4158-611: A Banach space. A Hilbert space H {\displaystyle H} on K = R , C {\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} } is complete for a norm of the form ‖ x ‖ H = ⟨ x , x ⟩ , {\displaystyle \|x\|_{H}={\sqrt {\langle x,x\rangle }},} where ⟨ ⋅ , ⋅ ⟩ : H × H → K {\displaystyle \langle \cdot ,\cdot \rangle :H\times H\to \mathbb {K} }
4389-405: A Banach space. In the case where I {\displaystyle I} is finite with n {\displaystyle n} elements, this construction yields R n {\displaystyle \mathbb {R} ^{n}} with the p {\displaystyle p} -norm defined above. If I {\displaystyle I} is countably infinite, this
4620-417: A Hilbert basis E , {\displaystyle E,} i.e., a maximal orthonormal subset of L 2 {\displaystyle L^{2}} or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to ℓ 2 ( E ) {\displaystyle \ell ^{2}(E)} (same E {\displaystyle E} as above), i.e.,
4851-466: A Hilbert space of type ℓ 2 . {\displaystyle \ell ^{2}.} The Euclidean length of a vector x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} in the n {\displaystyle n} -dimensional real vector space R n {\displaystyle \mathbb {R} ^{n}}
SECTION 20
#17327866355155082-490: A closed affine hyperplane . The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane. A subset S {\displaystyle S} in a Banach space X {\displaystyle X} is total if the linear span of S {\displaystyle S} is dense in X . {\displaystyle X.} The subset S {\displaystyle S}
5313-503: A compact Hausdorff space K , {\displaystyle K,} equipped with the max norm, ‖ f ‖ C ( K ) = max { | f ( x ) | : x ∈ K } , f ∈ C ( K ) . {\displaystyle \|f\|_{C(K)}=\max\{|f(x)|:x\in K\},\quad f\in C(K).} According to
5544-1232: A complex parameter z {\textstyle z} as follows: f z = ∑ j = 1 m | a j | u ( z ) u ( θ ) e i α j 1 A j g z = ∑ k = 1 n | b k | 1 − v ( z ) 1 − v ( θ ) e i β k 1 B k {\displaystyle {\begin{aligned}f_{z}&=\sum _{j=1}^{m}\left\vert a_{j}\right\vert ^{\frac {u(z)}{u(\theta )}}\mathrm {e} ^{\mathrm {i} \alpha _{j}}\mathbf {1} _{A_{j}}\\g_{z}&=\sum _{k=1}^{n}\left\vert b_{k}\right\vert ^{\frac {1-v(z)}{1-v(\theta )}}\mathrm {e} ^{\mathrm {i} \beta _{k}}\mathbf {1} _{B_{k}}\end{aligned}}} so that f θ = f {\textstyle f_{\theta }=f} and g θ = g {\textstyle g_{\theta }=g} . Here, we are implicitly excluding
5775-513: A complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, ( 1 , 1 , 1 , … ) , {\displaystyle (1,1,1,\ldots ),} will have an infinite p {\displaystyle p} -norm for 1 ≤ p < ∞ . {\displaystyle 1\leq p<\infty .} The space ℓ p {\displaystyle \ell ^{p}}
6006-435: A function in L p ( μ 1 ) {\textstyle L^{p}(\mu _{1})} . Then ‖ f ‖ p = sup | ∫ Ω 1 f g d μ 1 | {\displaystyle \lVert f\rVert _{p}=\sup {\biggl |}\int _{\Omega _{1}}fg\,\mathrm {d} \mu _{1}{\biggr |}} where
6237-461: A generalization of inner products , which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces. The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors . A normed space X {\displaystyle \ X\ }
6468-1043: A generic f {\textstyle f} ) ‖ T g − T g n ‖ p 0 ≤ ‖ T ‖ L p 0 → L q 0 ‖ g − g n ‖ p 0 → 0 ‖ T h − T h n ‖ p 1 ≤ ‖ T ‖ L p 1 → L q 1 ‖ h − h n ‖ p 1 → 0. {\displaystyle {\begin{aligned}\lVert Tg-Tg_{n}\rVert _{p_{0}}&\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}\lVert g-g_{n}\rVert _{p_{0}}\to 0&\lVert Th-Th_{n}\rVert _{p_{1}}&\leq \|T\|_{L^{p_{1}}\to L^{q_{1}}}\lVert h-h_{n}\rVert _{p_{1}}\to 0.\end{aligned}}} It
6699-688: A generic simple function, ‖ T f ‖ q θ ≤ ‖ T ‖ L p 0 → L q 0 1 − θ ‖ T ‖ L p 1 → L q 1 θ ‖ f ‖ p θ . {\displaystyle \lVert Tf\rVert _{q_{\theta }}\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}^{1-\theta }\|T\|_{L^{p_{1}}\to L^{q_{1}}}^{\theta }\lVert f\rVert _{p_{\theta }}.} Let us now prove that our claim ( 3 )
6930-440: A linear bijection T : X → Y {\displaystyle T:X\to Y} such that T {\displaystyle T} and its inverse T − 1 {\displaystyle T^{-1}} are continuous. If one of the two spaces X {\displaystyle X} or Y {\displaystyle Y} is complete (or reflexive , separable , etc.) then so
7161-564: A linear functional F : X → K {\displaystyle F:X\to \mathbb {K} } so that F | Y = f , and for all x ∈ X , Re ( F ( x ) ) ≤ p ( x ) . {\displaystyle F{\big \vert }_{Y}=f,\quad {\text{ and }}\quad {\text{ for all }}x\in X,\ \ \operatorname {Re} (F(x))\leq p(x).} In particular, every continuous linear functional on
Lp space - Misplaced Pages Continue
7392-671: A measurable function belongs to L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )} if and only if its absolute value does. Because of this, many formulas involving p {\displaystyle p} -norms are stated only for non-negative real-valued functions. Consider for example the identity ‖ f ‖ p r = ‖ f r ‖ p / r , {\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},} which holds whenever f ≥ 0 {\displaystyle f\geq 0}
7623-419: A measure of how much the two spaces X {\displaystyle X} and Y {\displaystyle Y} differ. Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function . So in particular, because
7854-1640: A natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex ) numbers are given by: ( x 1 , x 2 , … , x n , x n + 1 , … ) + ( y 1 , y 2 , … , y n , y n + 1 , … ) = ( x 1 + y 1 , x 2 + y 2 , … , x n + y n , x n + 1 + y n + 1 , … ) , λ ⋅ ( x 1 , x 2 , … , x n , x n + 1 , … ) = ( λ x 1 , λ x 2 , … , λ x n , λ x n + 1 , … ) . {\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}} Define
8085-436: A norm, the associated metric, known as Hamming distance , is a valid distance, since homogeneity is not required for distances. The p {\displaystyle p} -norm can be extended to vectors that have an infinite number of components ( sequences ), which yields the space ℓ p . {\displaystyle \ell ^{p}.} This contains as special cases: The space of sequences has
8316-455: A normed space ( X , ‖ ⋅ ‖ ) {\displaystyle \ (X,\|\cdot \|)\ } induces the usual metric topology τ d {\displaystyle \ \tau _{d}\ } on X , {\displaystyle \ X\ ,} which is referred to as the canonical or norm induced topology . Every normed space
8547-1188: A normed space ( X , ‖ ⋅ ‖ ) {\displaystyle \ (X,\|\cdot \|)\ } is called a complete norm if ( X , ‖ ⋅ ‖ ) {\displaystyle \ (X,\|\cdot \|)\ } is a Banach space. For any normed space ( X , ‖ ⋅ ‖ ) , {\displaystyle \ (X,\|\cdot \|)\ ,} there exists an L-semi-inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \ \langle \cdot ,\cdot \rangle \ } on X {\displaystyle \ X\ } such that ‖ x ‖ = ⟨ x , x ⟩ {\textstyle \ \|x\|={\sqrt {\langle x,x\rangle }}\ } for all x ∈ X ; {\displaystyle \ x\in X\ ;} in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are
8778-3552: A parallel argument, each ζ ∈ Ω 2 {\textstyle \zeta \in \Omega _{2}} belongs to (at most) one of the sets supporting g {\textstyle g} , say B k ^ {\textstyle B_{\hat {k}}} , and | g i y ( ζ ) | = | b k ^ | 1 − 1 / q 0 1 − 1 / q θ = | g ( ζ ) | 1 − 1 / q 0 1 − 1 / q θ = | g ( ζ ) | q θ ′ q 0 ′ ⟹ ‖ g i y ‖ q 0 ′ ≤ ‖ g ‖ q θ ′ q θ ′ q 0 ′ . {\displaystyle \left\vert g_{\mathrm {i} y}(\zeta )\right\vert =\left\vert b_{\hat {k}}\right\vert ^{\frac {1-1/q_{0}}{1-1/q_{\theta }}}=\left\vert g(\zeta )\right\vert ^{\frac {1-1/q_{0}}{1-1/q_{\theta }}}=\left\vert g(\zeta )\right\vert ^{\frac {q_{\theta }'}{q_{0}'}}\implies \lVert g_{\mathrm {i} y}\rVert _{q_{0}'}\leq \lVert g\rVert _{q_{\theta }'}^{\frac {q_{\theta }'}{q_{0}'}}.} We can now bound Φ ( i y ) {\textstyle \Phi (\mathrm {i} y)} : By applying Hölder’s inequality with conjugate exponents q 0 {\textstyle q_{0}} and q 0 ′ {\textstyle q_{0}'} , we have | Φ ( i y ) | ≤ ‖ T f i y ‖ q 0 ‖ g i y ‖ q 0 ′ ≤ ‖ T ‖ L p 0 → L q 0 ‖ f i y ‖ p 0 ‖ g i y ‖ q 0 ′ = ‖ T ‖ L p 0 → L q 0 ‖ f ‖ p θ p θ p 0 ‖ g ‖ q θ ′ q θ ′ q 0 ′ = ‖ T ‖ L p 0 → L q 0 . {\displaystyle {\begin{aligned}\left\vert \Phi (\mathrm {i} y)\right\vert &\leq \lVert Tf_{\mathrm {i} y}\rVert _{q_{0}}\lVert g_{\mathrm {i} y}\rVert _{q_{0}'}\\&\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}\lVert f_{\mathrm {i} y}\rVert _{p_{0}}\lVert g_{\mathrm {i} y}\rVert _{q_{0}'}\\&=\|T\|_{L^{p_{0}}\to L^{q_{0}}}\lVert f\rVert _{p_{\theta }}^{\frac {p_{\theta }}{p_{0}}}\lVert g\rVert _{q_{\theta }'}^{\frac {q_{\theta }'}{q_{0}'}}\\&=\|T\|_{L^{p_{0}}\to L^{q_{0}}}.\end{aligned}}} We can repeat
9009-457: A separable infinite–dimensional Banach (or Fréchet ) space can be topologically embedded as an open subset of ℓ 2 ( N ) {\displaystyle \ \ell ^{2}(\mathbb {N} )\ } and, consequently, also admits a unique smooth structure making it into a C ∞ {\displaystyle \ C^{\infty }\ } Hilbert manifold . There
9240-4383: A sequence of simple functions such that | f n | ≤ | f | {\textstyle \left\vert f_{n}\right\vert \leq \left\vert f\right\vert } , for all n {\textstyle n} , and f n → f {\textstyle f_{n}\to f} pointwise. Let E = { x ∈ Ω 1 : | f ( x ) | > 1 } {\textstyle E=\{x\in \Omega _{1}:\left\vert f(x)\right\vert >1\}} and define g = f 1 E {\textstyle g=f\mathbf {1} _{E}} , g n = f n 1 E {\textstyle g_{n}=f_{n}\mathbf {1} _{E}} , h = f − g = f 1 E c {\textstyle h=f-g=f\mathbf {1} _{E^{\mathrm {c} }}} and h n = f n − g n {\textstyle h_{n}=f_{n}-g_{n}} . Note that, since we are assuming p 0 ≤ p θ ≤ p 1 {\textstyle p_{0}\leq p_{\theta }\leq p_{1}} , ‖ f ‖ p θ p θ = ∫ Ω 1 | f | p θ d μ 1 ≥ ∫ Ω 1 | f | p θ 1 E d μ 1 ≥ ∫ Ω 1 | f 1 E | p 0 d μ 1 = ∫ Ω 1 | g | p 0 d μ 1 = ‖ g ‖ p 0 p 0 ‖ f ‖ p θ p θ = ∫ Ω 1 | f | p θ d μ 1 ≥ ∫ Ω 1 | f | p θ 1 E c d μ 1 ≥ ∫ Ω 1 | f 1 E c | p 1 d μ 1 = ∫ Ω 1 | h | p 1 d μ 1 = ‖ h ‖ p 1 p 1 {\displaystyle {\begin{aligned}\lVert f\rVert _{p_{\theta }}^{p_{\theta }}&=\int _{\Omega _{1}}\left\vert f\right\vert ^{p_{\theta }}\,\mathrm {d} \mu _{1}\geq \int _{\Omega _{1}}\left\vert f\right\vert ^{p_{\theta }}\mathbf {1} _{E}\,\mathrm {d} \mu _{1}\geq \int _{\Omega _{1}}\left\vert f\mathbf {1} _{E}\right\vert ^{p_{0}}\,\mathrm {d} \mu _{1}=\int _{\Omega _{1}}\left\vert g\right\vert ^{p_{0}}\,\mathrm {d} \mu _{1}=\lVert g\rVert _{p_{0}}^{p_{0}}\\\lVert f\rVert _{p_{\theta }}^{p_{\theta }}&=\int _{\Omega _{1}}\left\vert f\right\vert ^{p_{\theta }}\,\mathrm {d} \mu _{1}\geq \int _{\Omega _{1}}\left\vert f\right\vert ^{p_{\theta }}\mathbf {1} _{E^{\mathrm {c} }}\,\mathrm {d} \mu _{1}\geq \int _{\Omega _{1}}\left\vert f\mathbf {1} _{E^{\mathrm {c} }}\right\vert ^{p_{1}}\,\mathrm {d} \mu _{1}=\int _{\Omega _{1}}\left\vert h\right\vert ^{p_{1}}\,\mathrm {d} \mu _{1}=\lVert h\rVert _{p_{1}}^{p_{1}}\end{aligned}}} and, equivalently, g ∈ L p 0 ( Ω 1 ) {\textstyle g\in L^{p_{0}}(\Omega _{1})} and h ∈ L p 1 ( Ω 1 ) {\textstyle h\in L^{p_{1}}(\Omega _{1})} . Let us see what happens in
9471-2362: A simple function Ω 2 → C {\textstyle \Omega _{2}\to \mathbb {C} } , namely g = ∑ k = 1 n b k 1 B k {\displaystyle g=\sum _{k=1}^{n}b_{k}\mathbf {1} _{B_{k}}} for some finite n ∈ N {\textstyle n\in \mathbb {N} } , b k = | b k | e i β k ∈ C {\textstyle b_{k}=\left\vert b_{k}\right\vert \mathrm {e} ^{\mathrm {i} \beta _{k}}\in \mathbb {C} } and B k ∈ Σ 2 {\textstyle B_{k}\in \Sigma _{2}} , k = 1 , 2 , … , n {\textstyle k=1,2,\dots ,n} . Note that, since we are assuming Ω 1 {\textstyle \Omega _{1}} and Ω 2 {\textstyle \Omega _{2}} to be σ {\textstyle \sigma } -finite metric spaces, f ∈ L r ( μ 1 ) {\textstyle f\in L^{r}(\mu _{1})} and g ∈ L r ( μ 2 ) {\textstyle g\in L^{r}(\mu _{2})} for all r ∈ [ 1 , ∞ ] {\textstyle r\in [1,\infty ]} . Then, by proper normalization, we can assume ‖ f ‖ p θ = 1 {\textstyle \lVert f\rVert _{p_{\theta }}=1} and ‖ g ‖ q θ ′ = 1 {\textstyle \lVert g\rVert _{q_{\theta }'}=1} , with q θ ′ = q θ ( q θ − 1 ) − 1 {\textstyle q_{\theta }'=q_{\theta }(q_{\theta }-1)^{-1}} and with p θ {\textstyle p_{\theta }} , q θ {\textstyle q_{\theta }} as defined by
Lp space - Misplaced Pages Continue
9702-452: A solution's vector of parameter values (i.e. the sum of its absolute values), or its squared L 2 {\displaystyle L^{2}} norm (its Euclidean length ). Techniques which use an L1 penalty, like LASSO , encourage sparse solutions (where the many parameters are zero). Elastic net regularization uses a penalty term that is a combination of the L 1 {\displaystyle L^{1}} norm and
9933-508: A space is the Fréchet space C ∞ ( K ) , {\displaystyle C^{\infty }(K),} whose definition can be found in the article on spaces of test functions and distributions . Complete norms vs complete topological vector spaces There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses
10164-578: A space of measurable functions for which the p {\displaystyle p} -th power of the absolute value is Lebesgue integrable , where functions which agree almost everywhere are identified. More generally, let ( S , Σ , μ ) {\displaystyle (S,\Sigma ,\mu )} be a measure space and 1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty .} When p ≠ ∞ {\displaystyle p\neq \infty } , consider
10395-743: A subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional. An important special case is the following: for every vector x {\displaystyle x} in a normed space X , {\displaystyle X,} there exists a continuous linear functional f {\displaystyle f} on X {\displaystyle X} such that f ( x ) = ‖ x ‖ X , ‖ f ‖ X ′ ≤ 1. {\displaystyle f(x)=\|x\|_{X},\quad \|f\|_{X^{\prime }}\leq 1.} When x {\displaystyle x}
10626-1000: A sum is used to define the p {\displaystyle p} -norm. In complete analogy to the preceding definition one can define the space ℓ p ( I ) {\displaystyle \ell ^{p}(I)} over a general index set I {\displaystyle I} (and 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } ) as ℓ p ( I ) = { ( x i ) i ∈ I ∈ K I : ∑ i ∈ I | x i | p < + ∞ } , {\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<;+\infty \right\},} where convergence on
10857-419: A sum: f = g + h , where g ∈ L and h ∈ L . In particular, the above result implies that L is included in L + L , the sumset of L and L in the space of all measurable functions. Therefore, we have the following chain of inclusions: Corollary — L ∩ L ⊂ L ⊂ L + L . In practice, we often encounter operators defined on
11088-961: A union U = ⋃ x ∈ I B r x ( x ) = ⋃ x ∈ I x + B r x ( 0 ) = ⋃ x ∈ I x + r x B 1 ( 0 ) {\displaystyle \ U=\bigcup _{x\in I}B_{r_{x}}(x)=\bigcup _{x\in I}x+B_{r_{x}}(0)=\bigcup _{x\in I}x+r_{x}\ B_{1}(0)\ } indexed by some subset I ⊆ U , {\displaystyle \ I\subseteq U\ ,} where every r x {\displaystyle \ r_{x}\ } may be picked from
11319-407: A vector space X {\displaystyle \ X\ } then ( X , p ) {\displaystyle \ (X,p)\ } is a Banach space if and only if ( X , q ) {\displaystyle \ (X,q)\ } is a Banach space. See this footnote for an example of a continuous norm on a Banach space that
11550-410: Is not associated with any particular norm or metric (both of which are " forgotten "). This Hausdorff TVS ( X , τ d ) {\displaystyle \left(X,\tau _{d}\right)} is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS
11781-427: Is not equivalent to that Banach space's given norm. All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. A metric D {\displaystyle D} on a vector space X {\displaystyle X} is induced by a norm on X {\displaystyle X} if and only if D {\displaystyle D}
SECTION 50
#173278663551512012-612: Is not translation invariant, then it may be possible for ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} to be a Banach space but for ( X , D ) {\displaystyle (X,D)} to not be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces , implies that if there exists any complete metric D {\displaystyle D} on X {\displaystyle X} that induces
12243-937: Is finer or coarser than the other then they must be equal (that is, if τ ⊆ τ 2 {\displaystyle \tau \subseteq \tau _{2}} or τ 2 ⊆ τ {\displaystyle \ \tau _{2}\subseteq \tau \ } then τ = τ 2 {\displaystyle \ \tau =\tau _{2}\ } ). So for example, if ( X , p ) {\displaystyle \ (X,p)\ } and ( X , q ) {\displaystyle \ (X,q)\ } are Banach spaces with topologies τ p {\displaystyle \tau _{p}} and τ q {\displaystyle \ \tau _{q}\ } and if one of these spaces has some open ball that
12474-484: Is homeomorphic to the product space ∏ i ∈ N R {\textstyle \ \prod _{i\in \mathbb {N} }\mathbb {R} \ } of countably many copies of R {\displaystyle \ \mathbb {R} \ } (this homeomorphism need not be a linear map ). Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology
12705-471: Is normable if and only if its strong dual space X b ′ {\displaystyle X_{b}^{\prime }} is normable, in which case X b ′ {\displaystyle X_{b}^{\prime }} is a Banach space ( X b ′ {\displaystyle X_{b}^{\prime }} denotes the strong dual space of X , {\displaystyle X,} whose topology
12936-405: Is translation invariant and absolutely homogeneous , which means that D ( s x , s y ) = | s | D ( x , y ) {\displaystyle D(sx,sy)=|s|D(x,y)} for all scalars s {\displaystyle s} and all x , y ∈ X , {\displaystyle x,y\in X,} in which case
13167-463: Is translation invariant , which means that for any x ∈ X {\displaystyle \ x\in X\ } and S ⊆ X , {\displaystyle \ S\subseteq X\ ,} the subset S {\displaystyle \ S\ } is open (respectively, closed ) in X {\displaystyle \ X\ } if and only if this
13398-412: Is a separable space , which by definition means that X {\displaystyle \ X\ } contains some countable dense subset . All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to
13629-416: Is a Fréchet–Urysohn space . This shows that in the category of locally convex TVSs , Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces . Every normed space can be isometrically embedded onto a dense vector subspace of some Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion
13860-492: Is a Hilbert space , or to L and L . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps. First we need the following definition: By splitting up the function f in L as
14091-548: Is a closed linear subspace of a normed space X , {\displaystyle X,} there is a natural norm on the quotient space X / M , {\displaystyle X/M,} ‖ x + M ‖ = inf m ∈ M ‖ x + m ‖ . {\displaystyle \|x+M\|=\inf \limits _{m\in M}\|x+m\|.} The quotient X / M {\displaystyle X/M}
SECTION 60
#173278663551514322-568: Is a complete normed vector space . Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach , who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly . Maurice René Fréchet
14553-533: Is a convex and bounded subset of X , {\displaystyle \ X\ ,} but a compact ball / neighborhood exists if and only if X {\displaystyle \ X\ } is a finite-dimensional vector space . In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property . If x 0 {\displaystyle \ x_{0}\ }
14784-400: Is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ n p {\displaystyle \ell _{n}^{p}} is to denote by C p ( n ) {\displaystyle C_{p}(n)} the smallest constant C {\displaystyle C} such that
15015-576: Is a metrizable topological vector space (such as any norm induced topology, for example), then ( X , τ ) {\displaystyle (X,\tau )} is a complete TVS if and only if it is a sequentially complete TVS, meaning that it is enough to check that every Cauchy sequence in ( X , τ ) {\displaystyle (X,\tau )} converges in ( X , τ ) {\displaystyle (X,\tau )} to some point of X {\displaystyle X} (that is, there
15246-407: Is a Banach space A {\displaystyle A} over K = R {\displaystyle \mathbb {K} =\mathbb {R} } or C , {\displaystyle \mathbb {C} ,} together with a structure of algebra over K {\displaystyle \mathbb {K} } , such that the product map A × A ∋ (
15477-832: Is a Banach space if and only if each absolutely convergent series in X {\displaystyle \ X\ } converges to a value that lies within X , {\displaystyle \ X\ ,} ∑ n = 1 ∞ ‖ v n ‖ < ∞ implies that ∑ n = 1 ∞ v n converges in X . {\displaystyle \ \sum _{n=1}^{\infty }\|v_{n}\|<\infty \quad {\text{ implies that }}\quad \sum _{n=1}^{\infty }v_{n}\ \ {\text{ converges in }}\ \ X~.} The canonical metric d {\displaystyle d} of
15708-597: Is a Banach space when X {\displaystyle X} is complete. The quotient map from X {\displaystyle X} onto X / M , {\displaystyle X/M,} sending x ∈ X {\displaystyle x\in X} to its class x + M , {\displaystyle x+M,} is linear, onto and has norm 1 , {\displaystyle 1,} except when M = X , {\displaystyle M=X,} in which case
15939-412: Is a Banach space, the space B ( X ) = B ( X , X ) {\displaystyle B(X)=B(X,X)} forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps. If X {\displaystyle X} and Y {\displaystyle Y} are normed spaces, they are isomorphic normed spaces if there exists
16170-414: Is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences R N = ∏ i ∈ N R {\textstyle \ \mathbb {R} ^{\mathbb {N} }=\prod _{i\in \mathbb {N} }\mathbb {R} \ } with the product topology ). However,
16401-489: Is a closed linear subspace of the dual. The dual of M {\displaystyle M} is isometrically isomorphic to X ′ / M ⊥ . {\displaystyle X'/M^{\bot }.} The dual of X / M {\displaystyle X/M} is isometrically isomorphic to M ⊥ . {\displaystyle M^{\bot }.} Riesz%E2%80%93Thorin theorem In mathematics ,
16632-487: Is a compact subset S {\displaystyle \ S\ } of ℓ 2 ( N ) {\displaystyle \ \ell ^{2}(\mathbb {N} )\ } whose convex hull co ( S ) {\displaystyle \ \operatorname {co} (S)\ } is not closed and thus also not compact (see this footnote for an example). However, like in all Banach spaces,
16863-737: Is a consequence of Hölder's inequality . In R n {\displaystyle \mathbb {R} ^{n}} for n > 1 , {\displaystyle n>1,} the formula ‖ x ‖ p = ( | x 1 | p + | x 2 | p + ⋯ + | x n | p ) 1 / p {\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}} defines an absolutely homogeneous function for 0 < p < 1 ; {\displaystyle 0<p<1;} however,
17094-786: Is a consequence of the Riesz–Thorin interpolation theorem , and is made precise with the Hausdorff–Young inequality . By contrast, if p > 2 , {\displaystyle p>2,} the Fourier transform does not map into L q . {\displaystyle L^{q}.} Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus . The spaces L 2 {\displaystyle L^{2}} and ℓ 2 {\displaystyle \ell ^{2}} are both Hilbert spaces. In fact, by choosing
17325-456: Is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram. The interpolation theorem was originally stated and proved by Marcel Riesz in 1927. The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that p 0 ≤ q 0 and p 1 ≤ q 1 . Olof Thorin extended
17556-450: Is a generalization of the dual norm -induced topology on the continuous dual space X ′ {\displaystyle X^{\prime }} ; see this footnote for more details). If X {\displaystyle X} is a metrizable locally convex TVS, then X {\displaystyle X} is normable if and only if X b ′ {\displaystyle X_{b}^{\prime }}
17787-558: Is a measurable function that is equal to 0 {\displaystyle 0} almost everywhere then ‖ f ‖ p = 0 {\displaystyle \|f\|_{p}=0} for every p {\displaystyle p} and thus f ∈ L p ( S , μ ) {\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )} for all p . {\displaystyle p.} For every positive p , {\displaystyle p,}
18018-509: Is a norm if and only if no such f {\displaystyle f} exists). Zero sets of p {\displaystyle p} -seminorms If f {\displaystyle f} is measurable and equals 0 {\displaystyle 0} a.e. then ‖ f ‖ p = 0 {\displaystyle \|f\|_{p}=0} for all positive p ≤ ∞ . {\displaystyle p\leq \infty .} On
18249-419: Is a normed space and that τ {\displaystyle \tau } is the norm topology induced on X . {\displaystyle X.} Suppose that D {\displaystyle D} is any metric on X {\displaystyle X} such that the topology that D {\displaystyle D} induces on X {\displaystyle X}
18480-413: Is a pair ( X , ‖ ⋅ ‖ ) {\displaystyle \ (X,\|\cdot \|)\ } consisting of a vector space X {\displaystyle \ X\ } over a scalar field K {\displaystyle \mathbb {K} } (where K {\displaystyle \ \mathbb {K} \ }
18711-538: Is a rational number with an even numerator in its reduced form, and x {\displaystyle x} is drawn from the set of real numbers, or one of its subsets. The Euclidean norm from above falls into this class and is the 2 {\displaystyle 2} -norm, and the 1 {\displaystyle 1} -norm is the norm that corresponds to the rectilinear distance . The L ∞ {\displaystyle L^{\infty }} -norm or maximum norm (or uniform norm)
18942-688: Is a sequence in of positive real numbers that converges to 0 {\displaystyle \ 0\ } in R {\displaystyle \mathbb {R} } (such as r n := 1 n {\displaystyle \ r_{n}:={\tfrac {1}{n}}\ } or r n := 1 / 2 n , {\displaystyle \ r_{n}:=1/2^{n}\ ,} for instance). So for example, every open subset U {\displaystyle \ U\ } of X {\displaystyle \ X\ } can be written as
19173-759: Is a simple function. As already mentioned, the inequality holds true for all f ∈ L p θ ( Ω 1 ) {\textstyle f\in L^{p_{\theta }}(\Omega _{1})} by the density of simple functions in L p θ ( Ω 1 ) {\textstyle L^{p_{\theta }}(\Omega _{1})} . Formally, let f ∈ L p θ ( Ω 1 ) {\textstyle f\in L^{p_{\theta }}(\Omega _{1})} and let ( f n ) n {\textstyle (f_{n})_{n}} be
19404-887: Is a vector and s ≠ 0 {\displaystyle \ s\neq 0\ } is a scalar then x 0 + s B r ( x ) = B | s | r ( x 0 + s x ) and x 0 + s C r ( x ) = C | s | r ( x 0 + s x ) . {\displaystyle \ x_{0}+s\ B_{r}(x)=B_{|s|r}\!\left(\ x_{0}+s\ x\ \right)\qquad {\text{ and }}\qquad x_{0}+s\ C_{r}(x)=C_{|s|r}\!\left(\ x_{0}+s\ x\ \right)~.} Using s = 1 {\displaystyle \ s=1\ } shows that this norm-induced topology
19635-448: Is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS ( X , τ d ) {\displaystyle \ \left(X,\tau _{d}\right)\ } is only a vector space together with a certain type of topology; that is to say, when considered as a TVS, it
19866-426: Is a vector subspace of L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )} for every positive p ≤ ∞ . {\displaystyle p\leq \infty .} Quotient vector space Banach space In mathematics , more specifically in functional analysis , a Banach space (pronounced [ˈbanax] )
20097-427: Is also normable , which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm . Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin. All Banach spaces are barrelled spaces , which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that
20328-426: Is also a consequence of Minkowski's inequality ‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p {\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}} which establishes that ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} satisfies
20559-448: Is also an open subset of the other space (or equivalently, if one of p : ( X , τ q ) → R {\displaystyle \ p:\left(X,\tau _{q}\right)\to \mathbb {R} \ } or q : ( X , τ p ) → R {\displaystyle \ q:\left(X,\tau _{p}\right)\to \mathbb {R} \ }
20790-983: Is always a continuous function with respect to the topology that it induces. The open and closed balls of radius r > 0 {\displaystyle r>0} centered at a point x ∈ X {\displaystyle \ x\in X\ } are, respectively, the sets B r ( x ) := { z ∈ X : ‖ z − x ‖ < r } and C r ( x ) := { z ∈ X : ‖ z − x ‖ ≤ r } . {\displaystyle \ B_{r}(x):=\{\ z\in X:\|z-x\|<;r\ \}\qquad {\text{ and }}\qquad C_{r}(x):=\{\ z\in X:\|z-x\|\leq r\ \}~.} Any such ball
21021-2075: Is an entire function, bounded on the strip 0 ≤ R e z ≤ 1 {\textstyle 0\leq \operatorname {\mathbb {R} e} z\leq 1} . Then, in order to prove ( 2 ), we only need to show that for all f z {\textstyle f_{z}} and g z {\textstyle g_{z}} as constructed above. Indeed, if ( 3 ) holds true, by Hadamard three-lines theorem , | Φ ( θ + i 0 ) | = | ∫ Ω 2 ( T f ) g d μ 2 | ≤ ‖ T ‖ L p 0 → L q 0 1 − θ ‖ T ‖ L p 1 → L q 1 θ {\displaystyle \left\vert \Phi (\theta +\mathrm {i} 0)\right\vert ={\biggl \vert }\int _{\Omega _{2}}(Tf)g\,\mathrm {d} \mu _{2}{\biggr \vert }\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}^{1-\theta }\|T\|_{L^{p_{1}}\to L^{q_{1}}}^{\theta }} for all f {\textstyle f} and g {\textstyle g} . This means, by fixing f {\textstyle f} , that sup g | ∫ Ω 2 ( T f ) g d μ 2 | ≤ ‖ T ‖ L p 0 → L q 0 1 − θ ‖ T ‖ L p 1 → L q 1 θ {\displaystyle \sup _{g}{\biggl \vert }\int _{\Omega _{2}}(Tf)g\,\mathrm {d} \mu _{2}{\biggr \vert }\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}^{1-\theta }\|T\|_{L^{p_{1}}\to L^{q_{1}}}^{\theta }} where
21252-459: Is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space , although there exist normed spaces that are Baire but not Banach. The norm ‖ ⋅ ‖ : ( X , τ d ) → R {\displaystyle \ \|\,\cdot \,\|:\left(X,\tau _{d}\right)\to \mathbb {R} \ }
21483-1671: Is called a Banach space and the canonical metric d {\displaystyle \ d\ } is called a complete metric if ( X , d ) {\displaystyle \ (X,d)\ } is a complete metric space , which by definition means for every Cauchy sequence x 1 , x 2 , … {\displaystyle \ x_{1},x_{2},\ldots \ } in ( X , d ) , {\displaystyle \ (X,d)\ ,} there exists some x ∈ X {\displaystyle \ x\in X\ } such that lim n → ∞ x n = x in ( X , d ) {\displaystyle \lim _{n\to \infty }x_{n}=x\;{\text{ in }}(X,d)} where because ‖ x n − x ‖ = d ( x n , x ) , {\displaystyle \left\|x_{n}-x\right\|=d\left(x_{n},x\right),} this sequence's convergence to x {\displaystyle x} can equivalently be expressed as: lim n → ∞ ‖ x n − x ‖ = 0 in R . {\displaystyle \lim _{n\to \infty }\left\|x_{n}-x\right\|=0\;{\text{ in }}\mathbb {R} ~.} The norm ‖ ⋅ ‖ {\displaystyle \ \|\cdot \|\ } of
21714-547: Is closed under scalar multiplication is due to ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} being absolutely homogeneous , which means that ‖ s f ‖ p = | s | ‖ f ‖ p {\displaystyle \|sf\|_{p}=|s|\|f\|_{p}} for every scalar s {\displaystyle s} and every function f . {\displaystyle f.} Absolute homogeneity ,
21945-426: Is commonly R {\displaystyle \ \mathbb {R} \ } or C {\displaystyle \ \mathbb {C} \ } ) together with a distinguished norm ‖ ⋅ ‖ : X → R . {\displaystyle \ \|\cdot \|:X\to \mathbb {R} ~.} Like all norms, this norm induces
22176-406: Is complete, thus making it a Banach space . This Banach space is the L p {\displaystyle L^{p}} -space over { 1 , 2 , … , n } . {\displaystyle \{1,2,\ldots ,n\}.} The grid distance or rectilinear distance (sometimes called the " Manhattan distance ") between two points is never shorter than the length of
22407-1073: Is continuous) then their topologies are identical and their norms are equivalent . Two norms, p {\displaystyle \ p\ } and q , {\displaystyle q,} on a vector space X {\displaystyle \ X\ } are said to be equivalent if they induce the same topology; this happens if and only if there exist positive real numbers c , C > 0 {\displaystyle \ c,\ C>0\ } such that c q ( x ) ≤ p ( x ) ≤ C q ( x ) {\textstyle \ c\ q(x)\leq p(x)\leq C\ q(x)\ } for all x ∈ X . {\displaystyle \ x\in X~.} If p {\displaystyle \ p\ } and q {\displaystyle q} are two equivalent norms on
22638-544: Is continuous, which happens if and only if { x ∈ X : | f ( x ) | < 1 } {\displaystyle \{x\in X:|f(x)|<;1\}} is an open subset of X . {\displaystyle X.} And very importantly for applying the Hahn–Banach theorem , a linear functional f {\displaystyle f} is continuous if and only if this
22869-586: Is convergent for p > 1. {\displaystyle p>1.} One also defines the ∞ {\displaystyle \infty } -norm using the supremum : ‖ x ‖ ∞ = sup ( | x 1 | , | x 2 | , … , | x n | , | x n + 1 | , … ) {\displaystyle \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )} and
23100-402: Is denoted by B ( X , Y ) . {\displaystyle B(X,Y).} In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X {\displaystyle X} to another normed space is continuous if and only if it is bounded on the closed unit ball of X . {\displaystyle X.} Thus,
23331-496: Is dense in Y . {\displaystyle Y.} If Z {\displaystyle Z} is another Banach space such that there is an isometric isomorphism from X {\displaystyle X} onto a dense subset of Z , {\displaystyle Z,} then Z {\displaystyle Z} is isometrically isomorphic to Y . {\displaystyle Y.} This Banach space Y {\displaystyle Y}
23562-479: Is discussed by Stefan Rolewicz in Metric Linear Spaces . The ℓ 0 {\displaystyle \ell _{0}} -normed space is studied in functional analysis, probability theory, and harmonic analysis. Another function was called the ℓ 0 {\displaystyle \ell _{0}} "norm" by David Donoho —whose quotation marks warn that this function
23793-444: Is equal to τ . {\displaystyle \tau .} If D {\displaystyle D} is translation invariant then ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} is a Banach space if and only if ( X , D ) {\displaystyle (X,D)} is a complete metric space. If D {\displaystyle D}
24024-573: Is even homeomorphic to its own unit sphere { x ∈ ℓ 2 ( N ) : ‖ x ‖ 2 = 1 } , {\displaystyle \ \left\{\ x\in \ell ^{2}(\mathbb {N} ):\|x\|_{2}=1\ \right\}\ ,} which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane R 2 {\displaystyle \ \mathbb {R} ^{2}\ }
24255-439: Is exactly the sequence space ℓ p {\displaystyle \ell ^{p}} defined above. For uncountable sets I {\displaystyle I} this is a non- separable Banach space which can be seen as the locally convex direct limit of ℓ p {\displaystyle \ell ^{p}} -sequence spaces. For p = 2 , {\displaystyle p=2,}
24486-609: Is finite then the formula ‖ f ‖ p p = ‖ | f | p ‖ 1 {\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}} relates the p {\displaystyle p} -norm to the 1 {\displaystyle 1} -norm. Seminormed space of p {\displaystyle p} -th power integrable functions Each set of functions L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )} forms
24717-421: Is finite then this follows from the p = 1 {\displaystyle p=1} case and the formula ‖ f ‖ p p = ‖ | f | p ‖ 1 {\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}} mentioned above. Thus if p ≤ ∞ {\displaystyle p\leq \infty }
24948-408: Is finite, or the left-hand side is infinite. Thus, we will consider ℓ p {\displaystyle \ell ^{p}} spaces for 1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty .} The p {\displaystyle p} -norm thus defined on ℓ p {\displaystyle \ell ^{p}}
25179-509: Is given by the Euclidean norm : ‖ x ‖ 2 = ( x 1 2 + x 2 2 + ⋯ + x n 2 ) 1 / 2 . {\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.} The Euclidean distance between two points x {\displaystyle x} and y {\displaystyle y}
25410-438: Is indeed a norm, and ℓ p {\displaystyle \ell ^{p}} together with this norm is a Banach space . The fully general L p {\displaystyle L^{p}} space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with " arbitrarily many components "; in other words, functions . An integral instead of
25641-716: Is indeed certain. The sequence ( A j ) j = 1 m {\textstyle (A_{j})_{j=1}^{m}} consists of disjoint subsets in Σ 1 {\textstyle \Sigma _{1}} and, thus, each ξ ∈ Ω 1 {\textstyle \xi \in \Omega _{1}} belongs to (at most) one of them, say A ȷ ^ {\textstyle A_{\hat {\jmath }}} . Then, for z = i y {\textstyle z=\mathrm {i} y} , | f i y ( ξ ) | = | |
25872-781: Is measurable and has measure zero. Similarly, a measurable function f {\displaystyle f} (and its absolute value ) is bounded (or dominated ) almost everywhere by a real number C , {\displaystyle C,} written | f | ≤ C {\displaystyle |f|\leq C} a.e. , if the (necessarily) measurable set { s ∈ S : | f ( s ) | > C } {\displaystyle \{s\in S:|f(s)|>;C\}} has measure zero. The space L ∞ ( S , μ ) {\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}
26103-1079: Is measurable, r > 0 {\displaystyle r>0} is real, and 0 < p ≤ ∞ {\displaystyle 0<p\leq \infty } (here ∞ / r = def ∞ {\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty } when p = ∞ {\displaystyle p=\infty } ). The non-negativity requirement f ≥ 0 {\displaystyle f\geq 0} can be removed by substituting | f | {\displaystyle |f|} in for f , {\displaystyle f,} which gives ‖ | f | ‖ p r = ‖ | f | r ‖ p / r . {\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.} Note in particular that when p = r {\displaystyle p=r}
26334-498: Is no need to consider the more general notion of arbitrary Cauchy nets ). If ( X , τ ) {\displaystyle (X,\tau )} is a topological vector space whose topology is induced by some (possibly unknown) norm (such spaces are called normable ), then ( X , τ ) {\displaystyle (X,\tau )} is a complete topological vector space if and only if X {\displaystyle X} may be assigned
26565-439: Is not a norm because it is not homogeneous . For example, scaling the vector x {\displaystyle x} by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing , information theory , and statistics –notably in compressed sensing in signal processing and computational harmonic analysis . Despite not being
26796-625: Is not a proper norm—is the number of non-zero entries of the vector x . {\displaystyle x.} Many authors abuse terminology by omitting the quotation marks. Defining 0 0 = 0 , {\displaystyle 0^{0}=0,} the zero "norm" of x {\displaystyle x} is equal to | x 1 | 0 + | x 2 | 0 + ⋯ + | x n | 0 . {\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.} This
27027-412: Is not equal to the 0 {\displaystyle \mathbf {0} } vector, the functional f {\displaystyle f} must have norm one, and is called a norming functional for x . {\displaystyle x.} The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by
27258-539: Is not homeomorphic to the unit circle , for instance). This pattern in homeomorphism classes extends to generalizations of metrizable ( locally Euclidean ) topological manifolds known as metric Banach manifolds , which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly). For example, every open subset U {\displaystyle \ U\ } of
27489-679: Is not in ℓ 1 , {\displaystyle \ell ^{1},} but it is in ℓ p {\displaystyle \ell ^{p}} for p > 1 , {\displaystyle p>1,} as the series 1 p + 1 2 p + ⋯ + 1 n p + 1 ( n + 1 ) p + ⋯ , {\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,} diverges for p = 1 {\displaystyle p=1} (the harmonic series ), but
27720-2476: Is now easy to prove that T g n → T g {\textstyle Tg_{n}\to Tg} and T h n → T h {\textstyle Th_{n}\to Th} in measure: For any ϵ > 0 {\textstyle \epsilon >0} , Chebyshev’s inequality yields μ 2 ( y ∈ Ω 2 : | T g − T g n | > ϵ ) ≤ ‖ T g − T g n ‖ q 0 q 0 ϵ q 0 {\displaystyle \mu _{2}(y\in \Omega _{2}:\left\vert Tg-Tg_{n}\right\vert >\epsilon )\leq {\frac {\lVert Tg-Tg_{n}\rVert _{q_{0}}^{q_{0}}}{\epsilon ^{q_{0}}}}} and similarly for T h − T h n {\textstyle Th-Th_{n}} . Then, T g n → T g {\textstyle Tg_{n}\to Tg} and T h n → T h {\textstyle Th_{n}\to Th} a.e. for some subsequence and, in turn, T f n → T f {\textstyle Tf_{n}\to Tf} a.e. Then, by Fatou’s lemma and recalling that ( 4 ) holds true for simple functions, ‖ T f ‖ q θ ≤ lim inf n → ∞ ‖ T f n ‖ q θ ≤ ‖ T ‖ L p θ → L q θ lim inf n → ∞ ‖ f n ‖ p θ = ‖ T ‖ L p θ → L q θ ‖ f ‖ p θ . {\displaystyle \lVert Tf\rVert _{q_{\theta }}\leq \liminf _{n\to \infty }\lVert Tf_{n}\rVert _{q_{\theta }}\leq \|T\|_{L^{p_{\theta }}\to L^{q_{\theta }}}\liminf _{n\to \infty }\lVert f_{n}\rVert _{p_{\theta }}=\|T\|_{L^{p_{\theta }}\to L^{q_{\theta }}}\lVert f\rVert _{p_{\theta }}.} The proof outline presented in
27951-417: Is of type ( p θ , q θ ) for all 0 < θ < 1 . In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of T is the collection of all points ( 1 / p , 1 / q ) in the unit square [0, 1] × [0, 1] such that T is of type ( p , q ) . The interpolation theorem states that the Riesz diagram of T
28182-584: Is positive and f {\displaystyle f} is any measurable function, then ‖ f ‖ p = 0 {\displaystyle \|f\|_{p}=0} if and only if f = 0 {\displaystyle f=0} almost everywhere . Since the right hand side ( f = 0 {\displaystyle f=0} a.e.) does not mention p , {\displaystyle p,} it follows that all ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} have
28413-504: Is sometimes denoted by X ^ . {\displaystyle {\widehat {X}}.} If X {\displaystyle X} and Y {\displaystyle Y} are normed spaces over the same ground field K , {\displaystyle \mathbb {K} ,} the set of all continuous K {\displaystyle \mathbb {K} } -linear maps T : X → Y {\displaystyle T:X\to Y}
28644-489: Is the Hausdorff–Young inequality . The Hausdorff–Young inequality can also be established for the Fourier transform on locally compact Abelian groups . The norm estimate of 1 is not optimal. See the main article for references. Let f be a fixed integrable function and let T be the operator of convolution with f , i.e., for each function g we have Tg = f ∗ g . It
28875-888: Is the inner product , linear in its first argument that satisfies the following: ⟨ y , x ⟩ = ⟨ x , y ⟩ ¯ , for all x , y ∈ H ⟨ x , x ⟩ ≥ 0 , for all x ∈ H ⟨ x , x ⟩ = 0 if and only if x = 0. {\displaystyle {\begin{aligned}\langle y,x\rangle &={\overline {\langle x,y\rangle }},\quad {\text{ for all }}x,y\in H\\\langle x,x\rangle &\geq 0,\quad {\text{ for all }}x\in H\\\langle x,x\rangle =0{\text{ if and only if }}x&=0.\end{aligned}}} For example,
29106-538: Is the Hausdorff completion of the normed space X . {\displaystyle X.} The underlying metric space for Y {\displaystyle Y} is the same as the metric completion of X , {\displaystyle X,} with the vector space operations extended from X {\displaystyle X} to Y . {\displaystyle Y.} The completion of X {\displaystyle X}
29337-510: Is the length ‖ x − y ‖ 2 {\displaystyle \|x-y\|_{2}} of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of
29568-679: Is the limit of the L p {\displaystyle L^{p}} -norms for p → ∞ . {\displaystyle p\to \infty .} It turns out that this limit is equivalent to the following definition: ‖ x ‖ ∞ = max { | x 1 | , | x 2 | , … , | x n | } {\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}} See L -infinity . For all p ≥ 1 , {\displaystyle p\geq 1,}
29799-754: Is the other space. Two normed spaces X {\displaystyle X} and Y {\displaystyle Y} are isometrically isomorphic if in addition, T {\displaystyle T} is an isometry , that is, ‖ T ( x ) ‖ = ‖ x ‖ {\displaystyle \|T(x)\|=\|x\|} for every x {\displaystyle x} in X . {\displaystyle X.} The Banach–Mazur distance d ( X , Y ) {\displaystyle d(X,Y)} between two isomorphic but not isometric spaces X {\displaystyle X} and Y {\displaystyle Y} gives
30030-416: Is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces. If f : X → R {\displaystyle f:X\to \mathbb {R} } is a subadditive function (such as a norm, a sublinear function , or real linear functional), then f {\displaystyle f} is continuous at
30261-627: Is the same as the essential supremum of the absolute value of f {\displaystyle f} : ‖ f ‖ ∞ = { esssup | f | if μ ( S ) > 0 , 0 if μ ( S ) = 0. {\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}} For example, if f {\displaystyle f}
30492-920: Is the set of all measurable functions f {\displaystyle f} that are bounded almost everywhere (by some real C {\displaystyle C} ) and ‖ f ‖ ∞ {\displaystyle \|f\|_{\infty }} is defined as the infimum of these bounds: ‖ f ‖ ∞ = def inf { C ∈ R ≥ 0 : | f ( s ) | ≤ C for almost every s } . {\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.} When μ ( S ) ≠ 0 {\displaystyle \mu (S)\neq 0} then this
30723-611: Is then defined as the set of all infinite sequences of real (or complex) numbers such that the p {\displaystyle p} -norm is finite. One can check that as p {\displaystyle p} increases, the set ℓ p {\displaystyle \ell ^{p}} grows larger. For example, the sequence ( 1 , 1 2 , … , 1 n , 1 n + 1 , … ) {\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}
30954-876: Is therefore natural to investigate the behavior of such operators on the intermediate subspaces L . To this end, we go back to our example and note that the Fourier transform on the sumset L + L was obtained by taking the sum of two instantiations of the same operator, namely F L 1 : L 1 ( R d ) → L ∞ ( R d ) , {\displaystyle {\mathcal {F}}_{L^{1}}:L^{1}(\mathbf {R} ^{d})\to L^{\infty }(\mathbf {R} ^{d}),} F L 2 : L 2 ( R d ) → L 2 ( R d ) . {\displaystyle {\mathcal {F}}_{L^{2}}:L^{2}(\mathbf {R} ^{d})\to L^{2}(\mathbf {R} ^{d}).} These really are
31185-569: Is total in X {\displaystyle X} if and only if the only continuous linear functional that vanishes on S {\displaystyle S} is the 0 {\displaystyle \mathbf {0} } functional: this equivalence follows from the Hahn–Banach theorem. If X {\displaystyle X} is the direct sum of two closed linear subspaces M {\displaystyle M} and N , {\displaystyle N,} then
31416-464: Is true of its real part Re f {\displaystyle \operatorname {Re} f} and moreover, ‖ Re f ‖ = ‖ f ‖ {\displaystyle \|\operatorname {Re} f\|=\|f\|} and the real part Re f {\displaystyle \operatorname {Re} f} completely determines f , {\displaystyle f,} which
31647-1301: Is true of its translation x + S := { x + s : s ∈ S } . {\displaystyle \ x+S:=\{\ x+s:s\in S\ \}~.} Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include: { B r ( 0 ) : r > 0 } , { C r ( 0 ) : r > 0 } , { B r n ( 0 ) : n ∈ N } , or { C r n ( 0 ) : n ∈ N } {\displaystyle \ \left\{\ B_{r}(0):r>0\ \right\},\qquad \left\{\ C_{r}(0):r>0\ \right\},\qquad \left\{\ B_{r_{n}}(0):n\in \mathbb {N} \ \right\},\qquad {\text{ or }}\qquad \left\{\ C_{r_{n}}(0):n\in \mathbb {N} \ \right\}\ } where r 1 , r 2 , … {\displaystyle \ r_{1},r_{2},\ldots \ }
31878-433: Is unique up to a homeomorphism). Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including ℓ 2 ( N ) . {\displaystyle \ \ell ^{2}(\mathbb {N} )~.} In fact, ℓ 2 ( N ) {\displaystyle \ \ell ^{2}(\mathbb {N} )\ }
32109-443: Is unique up to isometric isomorphism. More precisely, for every normed space X , {\displaystyle X,} there exist a Banach space Y {\displaystyle Y} and a mapping T : X → Y {\displaystyle T:X\to Y} such that T {\displaystyle T} is an isometric mapping and T ( X ) {\displaystyle T(X)}
32340-428: Is well defined because all elements in the same class have the same image. The mapping T 1 {\displaystyle T_{1}} is a linear bijection from X / ker T {\displaystyle X/\ker T} onto the range T ( X ) , {\displaystyle T(X),} whose inverse need not be bounded. Basic examples of Banach spaces include:
32571-485: Is well known that T is bounded from L to L and it is trivial that it is bounded from L to L (both bounds are by || f || 1 ). Therefore the Riesz–Thorin theorem gives ‖ f ∗ g ‖ p ≤ ‖ f ‖ 1 ‖ g ‖ p . {\displaystyle \|f*g\|_{p}\leq \|f\|_{1}\|g\|_{p}.} We take this inequality and switch
32802-617: Is why the Hahn–Banach theorem is often stated only for real linear functionals. Also, a linear functional f {\displaystyle f} on X {\displaystyle X} is continuous if and only if the seminorm | f | {\displaystyle |f|} is continuous, which happens if and only if there exists a continuous seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } such that | f | ≤ p {\displaystyle |f|\leq p} ; this last statement involving
33033-584: The ℓ 0 {\displaystyle \ell _{0}} norm was established by Banach 's Theory of Linear Operations . The space of sequences has a complete metric topology provided by the F-norm ( x n ) ↦ ∑ n 2 − n | x n | 1 + | x n | , {\displaystyle (x_{n})\mapsto \sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}},} which
33264-660: The p {\displaystyle p} -norm or L p {\displaystyle L^{p}} -norm of x {\displaystyle x} is defined by ‖ x ‖ p = ( | x 1 | p + | x 2 | p + ⋯ + | x n | p ) 1 / p . {\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.} The absolute value bars can be dropped when p {\displaystyle p}
33495-778: The ‖ ⋅ ‖ 2 {\displaystyle \|\,\cdot \,\|_{2}} -norm is even induced by a canonical inner product ⟨ ⋅ , ⋅ ⟩ , {\displaystyle \langle \,\cdot ,\,\cdot \rangle ,} called the Euclidean inner product , which means that ‖ x ‖ 2 = ⟨ x , x ⟩ {\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} holds for all vectors x . {\displaystyle \mathbf {x} .} This inner product can expressed in terms of
33726-414: The 2 {\displaystyle 2} -norm is known: ‖ x ‖ 1 ≤ n ‖ x ‖ 2 . {\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.} This inequality depends on the dimension n {\displaystyle n} of the underlying vector space and follows directly from
33957-399: The p {\displaystyle p} -norm ‖ x ‖ p {\displaystyle \|x\|_{p}} of any given vector x {\displaystyle x} does not grow with p {\displaystyle p} : For the opposite direction, the following relation between the 1 {\displaystyle 1} -norm and
34188-547: The p {\displaystyle p} -norm: ‖ x ‖ p = ( | x 1 | p + | x 2 | p + ⋯ + | x n | p + | x n + 1 | p + ⋯ ) 1 / p {\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}} Here,
34419-424: The p {\displaystyle p} -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm ), which are that: Abstractly speaking, this means that R n {\displaystyle \mathbb {R} ^{n}} together with the p {\displaystyle p} -norm is a normed vector space . Moreover, it turns out that this space
34650-532: The p {\displaystyle p} -th power has a finite integral, or in symbols: ‖ f ‖ p = def ( ∫ S | f | p d μ ) 1 / p < ∞ . {\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .} To define
34881-539: The closed convex hull co ¯ S {\displaystyle \ {\overline {\operatorname {co} }}S\ } of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general not guaranteed that co ¯ S {\displaystyle \ {\overline {\operatorname {co} }}S\ } will be compact whenever S {\displaystyle S} is; an example can even be found in
35112-623: The orthogonal of M {\displaystyle M} in the dual, M ⊥ = { x ′ ∈ X : x ′ ( m ) = 0 , for all m ∈ M } . {\displaystyle M^{\bot }=\left\{x^{\prime }\in X:x^{\prime }(m)=0,\ {\text{ for all }}m\in M\right\}.} The orthogonal M ⊥ {\displaystyle M^{\bot }}
35343-625: The Banach–Mazur theorem , every Banach space is isometrically isomorphic to a subspace of some C ( K ) . {\displaystyle C(K).} For every separable Banach space X , {\displaystyle X,} there is a closed subspace M {\displaystyle M} of ℓ 1 {\displaystyle \ell ^{1}} such that X := ℓ 1 / M . {\displaystyle X:=\ell ^{1}/M.} Any Hilbert space serves as an example of
35574-731: The Banach–Steinhaus theorem holds. The open mapping theorem implies that if τ {\displaystyle \tau } and τ 2 {\displaystyle \ \tau _{2}\ } are topologies on X {\displaystyle \ X\ } that make both ( X , τ ) {\displaystyle \ (X,\tau )\ } and ( X , τ 2 ) {\displaystyle \ (X,\tau _{2})\ } into complete metrizable TVS (for example, Banach or Fréchet spaces ) and if one topology
35805-527: The Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier: H f ^ ( ξ ) = − i sgn ( ξ ) f ^ ( ξ ) . {\displaystyle {\widehat {{\mathcal {H}}f}}(\xi )=-i\,\operatorname {sgn}(\xi ){\hat {f}}(\xi ).} It follows from
36036-520: The Cauchy–Schwarz inequality to show that the Hilbert transform maps L ( R ) boundedly into itself for all n ≥ 2 . Interpolation now establishes the bound ‖ H f ‖ p ≤ A p ‖ f ‖ p {\displaystyle \|{\mathcal {H}}f\|_{p}\leq A_{p}\|f\|_{p}} for all 2 ≤ p < ∞ , and
36267-586: The Cauchy–Schwarz inequality . In general, for vectors in C n {\displaystyle \mathbb {C} ^{n}} where 0 < r < p : {\displaystyle 0<r<p:} ‖ x ‖ p ≤ ‖ x ‖ r ≤ n 1 r − 1 p ‖ x ‖ p . {\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.} This
36498-507: The Lp spaces L p {\displaystyle L^{p}} and their special cases, the sequence spaces ℓ p {\displaystyle \ell ^{p}} that consist of scalar sequences indexed by natural numbers N {\displaystyle \mathbb {N} } ; among them, the space ℓ 1 {\displaystyle \ell ^{1}} of absolutely summable sequences and
36729-591: The Marcinkiewicz interpolation theorem are better-suited for them. Furthermore, a good number of important operators, such as the Hardy-Littlewood maximal operator , are only sublinear . This is not a hindrance to applying real interpolation methods, but complex interpolation methods are ill-equipped to handle non-linear operators. On the other hand, real interpolation methods, compared to complex interpolation methods, tend to produce worse estimates on
36960-852: The Plancherel theorem that the Hilbert transform maps L ( R ) boundedly into itself. Nevertheless, the Hilbert transform is not bounded on L ( R ) or L ( R ) , and so we cannot use the Riesz–Thorin interpolation theorem directly. To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions 1 (−1,1) ( x ) and 1 (0,1) ( x ) − 1 (0,1) (− x ) . We can show, however, that ( H f ) 2 = f 2 + 2 H ( f H f ) {\displaystyle ({\mathcal {H}}f)^{2}=f^{2}+2{\mathcal {H}}(f{\mathcal {H}}f)} for all Schwartz functions f : R → C , and this identity can be used in conjunction with
37191-465: The Riesz–Thorin theorem , often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem , is a result about interpolation of operators . It is named after Marcel Riesz and his student G. Olof Thorin . This theorem bounds the norms of linear maps acting between L spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L which
37422-454: The coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product X × Y {\displaystyle X\times Y} (or the direct sum X ⊕ Y {\displaystyle X\oplus Y} ) is complete if and only if the two factors are complete. If M {\displaystyle M}
37653-403: The mean , median , and standard deviation , can be defined in terms of L p {\displaystyle L^{p}} metrics, and measures of central tendency can be characterized as solutions to variational problems . In penalized regression , "L1 penalty" and "L2 penalty" refer to penalizing either the L 1 {\displaystyle L^{1}} norm of
37884-404: The rectilinear distance , which takes into account that streets are either orthogonal or parallel to each other. The class of p {\displaystyle p} -norms generalizes these two examples and has an abundance of applications in many parts of mathematics , physics , and computer science . For a real number p ≥ 1 , {\displaystyle p\geq 1,}
38115-505: The same operator, in the sense that they agree on the subspace ( L ∩ L ) ( R ) . Since the intersection contains simple functions , it is dense in both L ( R ) and L ( R ) . Densely defined continuous operators admit unique extensions, and so we are justified in considering F L 1 {\displaystyle {\mathcal {F}}_{L^{1}}} and F L 2 {\displaystyle {\mathcal {F}}_{L^{2}}} to be
38346-434: The self-adjointness of the Hilbert transform can be used to carry over these bounds to the 1 < p ≤ 2 case. While the Riesz–Thorin interpolation theorem and its variants are powerful tools that yield a clean estimate on the interpolated operator norms, they suffer from numerous defects: some minor, some more severe. Note first that the complex-analytic nature of the proof of the Riesz–Thorin interpolation theorem forces
38577-878: The sumset L + L . For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps L ( R ) boundedly into L ( R ) , and Plancherel's theorem shows that the Fourier transform maps L ( R ) boundedly into itself, hence the Fourier transform F {\displaystyle {\mathcal {F}}} extends to ( L + L ) ( R ) by setting F ( f 1 + f 2 ) = F L 1 ( f 1 ) + F L 2 ( f 2 ) {\displaystyle {\mathcal {F}}(f_{1}+f_{2})={\mathcal {F}}_{L^{1}}(f_{1})+{\mathcal {F}}_{L^{2}}(f_{2})} for all f 1 ∈ L ( R ) and f 2 ∈ L ( R ) . It
38808-391: The triangle inequality for 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } (the triangle inequality does not hold for 0 < p < 1 {\displaystyle 0<p<1} ). That L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}
39039-450: The triangle inequality , and non-negativity are the defining properties of a seminorm . Thus ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} is a seminorm and the set L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )} of p {\displaystyle p} -th power integrable functions together with
39270-488: The Fourier coefficients f ^ ( n ) = 1 2 π ∫ − π π f ( x ) e − i n x d x , {\displaystyle {\hat {f}}(n)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx,} maps L ( T ) boundedly into ℓ ( Z ) and L ( T ) into ℓ ( Z ) . The Riesz–Thorin interpolation theorem now implies
39501-716: The Hardy space H ( R ) and the space BMO of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein . It has been shown in the first section that the Fourier transform F {\displaystyle {\mathcal {F}}} maps L ( R ) boundedly into L ( R ) and L ( R ) into itself. A similar argument shows that the Fourier series operator , which transforms periodic functions f : T → C into functions f ^ : Z → C {\displaystyle {\hat {f}}:\mathbf {Z} \to \mathbf {C} } whose values are
39732-453: The above section readily generalizes to the case in which the operator T is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function φ ( z ) = ∫ ( T z f z ) g z d μ 2 , {\displaystyle \varphi (z)=\int (T_{z}f_{z})g_{z}\,d\mu _{2},} from which we obtain
39963-638: The aforementioned sequence r 1 , r 2 , … {\displaystyle \ r_{1},r_{2},\ldots \ } (the open balls can be replaced with closed balls, although then the indexing set I {\displaystyle \ I\ } and radii r x {\displaystyle \ r_{x}\ } may also need to be replaced). Additionally, I {\displaystyle \ I\ } can always be chosen to be countable if X {\displaystyle \ X\ }
40194-449: The case q 0 = q 1 = 1 {\textstyle q_{0}=q_{1}=1} , which yields v ≡ 1 {\textstyle v\equiv 1} : In that case, one can simply take g z = g {\textstyle g_{z}=g} , independently of z {\textstyle z} , and the following argument will only require minor adaptations. Let us now introduce
40425-1336: The case p = ∞ . {\displaystyle p=\infty .} Define ℓ ∞ ( I ) = { x ∈ K I : sup range | x | < + ∞ } , {\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},} where for all x {\displaystyle x} ‖ x ‖ ∞ ≡ inf { C ∈ R ≥ 0 : | x i | ≤ C for all i ∈ I } = { sup range | x | if X ≠ ∅ , 0 if X = ∅ . {\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \varnothing ,\\0&{\text{if }}X=\varnothing .\end{cases}}} The index set I {\displaystyle I} can be turned into
40656-706: The case of the Hilbert transform in the previous section, we were able to bypass this problem by explicitly computing the norm estimates at several midway points. This is cumbersome and is often not possible in more general scenarios. Since many such operators satisfy the weak-type estimates μ ( { x : T f ( x ) > α } ) ≤ ( C p , q ‖ f ‖ p α ) q , {\displaystyle \mu \left(\{x:Tf(x)>\alpha \}\right)\leq \left({\frac {C_{p,q}\|f\|_{p}}{\alpha }}\right)^{q},} real interpolation theorems such as
40887-871: The connection between p , r and s is 1 r + 1 p = 1 + 1 s . {\displaystyle {\frac {1}{r}}+{\frac {1}{p}}=1+{\frac {1}{s}}.} The Hilbert transform of f : R → C is given by H f ( x ) = 1 π p . v . ∫ − ∞ ∞ f ( x − t ) t d t = ( 1 π p . v . 1 t ∗ f ) ( x ) , {\displaystyle {\mathcal {H}}f(x)={\frac {1}{\pi }}\,\mathrm {p.v.} \int _{-\infty }^{\infty }{\frac {f(x-t)}{t}}\,dt=\left({\frac {1}{\pi }}\,\mathrm {p.v.} {\frac {1}{t}}\ast f\right)(x),} where p.v. indicates
41118-517: The continuous dual is X ′ = B ( X , K ) {\displaystyle X^{\prime }=B(X,\mathbb {K} )} in this article. Since K {\displaystyle \mathbb {K} } is a Banach space (using the absolute value as norm), the dual X ′ {\displaystyle X^{\prime }} is a Banach space, for every normed space X . {\displaystyle X.} The Dixmier–Ng theorem characterizes
41349-527: The corresponding intermediate space L . There are several ways to state the Riesz–Thorin interpolation theorem; to be consistent with the notations in the previous section, we shall use the sumset formulation. Riesz–Thorin interpolation theorem — Let (Ω 1 , Σ 1 , μ 1 ) and (Ω 2 , Σ 2 , μ 2 ) be σ -finite measure spaces. Suppose 1 ≤ p 0 , q 0 , p 1 , q 1 ≤ ∞ , and let T : L ( μ 1 ) + L ( μ 1 ) → L ( μ 2 ) + L ( μ 2 ) be
41580-410: The corresponding space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded sequences. It turns out that ‖ x ‖ ∞ = lim p → ∞ ‖ x ‖ p {\displaystyle \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}} if the right-hand side
41811-444: The cost of losing absolute homogeneity. It does define an F-norm , though, which is homogeneous of degree p . {\displaystyle p.} Hence, the function d p ( x , y ) = ∑ i = 1 n | x i − y i | p {\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}} defines
42042-1979: The dominated convergence theorem one readily has ‖ f n ‖ p θ → ‖ f ‖ p θ ‖ g n ‖ p 0 → ‖ g ‖ p 0 ‖ h n ‖ p 1 → ‖ h ‖ p 1 . {\displaystyle {\begin{aligned}\lVert f_{n}\rVert _{p_{\theta }}&\to \lVert f\rVert _{p_{\theta }}&\lVert g_{n}\rVert _{p_{0}}&\to \lVert g\rVert _{p_{0}}&\lVert h_{n}\rVert _{p_{1}}&\to \lVert h\rVert _{p_{1}}.\end{aligned}}} Similarly, | f − f n | ≤ 2 | f | {\textstyle \left\vert f-f_{n}\right\vert \leq 2\left\vert f\right\vert } , | g − g n | ≤ 2 | g | {\textstyle \left\vert g-g_{n}\right\vert \leq 2\left\vert g\right\vert } and | h − h n | ≤ 2 | h | {\textstyle \left\vert h-h_{n}\right\vert \leq 2\left\vert h\right\vert } imply ‖ f − f n ‖ p θ → 0 ‖ g − g n ‖ p 0 → 0 ‖ h − h n ‖ p 1 → 0 {\displaystyle {\begin{aligned}\lVert f-f_{n}\rVert _{p_{\theta }}&\to 0&\lVert g-g_{n}\rVert _{p_{0}}&\to 0&\lVert h-h_{n}\rVert _{p_{1}}&\to 0\end{aligned}}} and, by
42273-436: The dual X ′ {\displaystyle X^{\prime }} of X {\displaystyle X} is isomorphic to the direct sum of the duals of M {\displaystyle M} and N . {\displaystyle N.} If M {\displaystyle M} is a closed linear subspace in X , {\displaystyle X,} one can associate
42504-423: The dual spaces of Banach spaces. The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem . Hahn–Banach theorem — Let X {\displaystyle X} be a vector space over the field K = R , C . {\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} .} Let further Then, there exists
42735-427: The first map π {\displaystyle \pi } is the quotient map, and the second map T 1 {\displaystyle T_{1}} sends every class x + ker T {\displaystyle x+\ker T} in the quotient to the image T ( x ) {\displaystyle T(x)} in Y . {\displaystyle Y.} This
42966-436: The following theorem of Elias Stein , published in his 1956 thesis: Stein interpolation theorem — Let (Ω 1 , Σ 1 , μ 1 ) and (Ω 2 , Σ 2 , μ 2 ) be σ -finite measure spaces. Suppose 1 ≤ p 0 , p 1 ≤ ∞, 1 ≤ q 0 , q 1 ≤ ∞ , and define: We take a collection of linear operators { T z : z ∈ S } on the space of simple functions in L ( μ 1 ) into
43197-819: The following: ‖ F f ‖ L q ( R d ) ≤ ‖ f ‖ L p ( R d ) ‖ f ^ ‖ ℓ q ( Z ) ≤ ‖ f ‖ L p ( T ) {\displaystyle {\begin{aligned}\left\|{\mathcal {F}}f\right\|_{L^{q}(\mathbf {R} ^{d})}&\leq \|f\|_{L^{p}(\mathbf {R} ^{d})}\\\left\|{\hat {f}}\right\|_{\ell ^{q}(\mathbf {Z} )}&\leq \|f\|_{L^{p}(\mathbf {T} )}\end{aligned}}} where 1 ≤ p ≤ 2 and 1 / p + 1 / q = 1 . This
43428-1461: The function Φ ( z ) = ∫ Ω 2 ( T f z ) g z d μ 2 = ∑ j = 1 m ∑ k = 1 n | a j | u ( z ) u ( θ ) | b k | 1 − v ( z ) 1 − v ( θ ) γ j , k {\displaystyle \Phi (z)=\int _{\Omega _{2}}(Tf_{z})g_{z}\,\mathrm {d} \mu _{2}=\sum _{j=1}^{m}\sum _{k=1}^{n}\left\vert a_{j}\right\vert ^{\frac {u(z)}{u(\theta )}}\left\vert b_{k}\right\vert ^{\frac {1-v(z)}{1-v(\theta )}}\gamma _{j,k}} where γ j , k = e i ( α j + β k ) ∫ Ω 2 ( T 1 A j ) 1 B k d μ 2 {\textstyle \gamma _{j,k}=\mathrm {e} ^{\mathrm {i} (\alpha _{j}+\beta _{k})}\int _{\Omega _{2}}(T\mathbf {1} _{A_{j}})\mathbf {1} _{B_{k}}\,\mathrm {d} \mu _{2}} are constants independent of z {\textstyle z} . We readily see that Φ ( z ) {\textstyle \Phi (z)}
43659-697: The function ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} defines a seminormed vector space . In general, the seminorm ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} is not a norm because there might exist measurable functions f {\displaystyle f} that satisfy ‖ f ‖ p = 0 {\displaystyle \|f\|_{p}=0} but are not identically equal to 0 {\displaystyle 0} ( ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}}
43890-505: The function ‖ x ‖ := D ( x , 0 ) {\displaystyle \|x\|:=D(x,0)} defines a norm on X {\displaystyle X} and the canonical metric induced by ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is equal to D . {\displaystyle D.} Suppose that ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)}
44121-424: The infinite-dimensional sequence space ℓ p {\displaystyle \ell ^{p}} defined below, is no longer locally convex. There is one ℓ 0 {\displaystyle \ell _{0}} norm and another function called the ℓ 0 {\displaystyle \ell _{0}} "norm" (with quotation marks). The mathematical definition of
44352-650: The interpolation theorem to the entire square, removing the lower-triangle restriction. The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis. We will first prove the result for simple functions and eventually show how the argument can be extended by density to all measurable functions. By symmetry, let us assume p 0 < p 1 {\textstyle p_{0}<p_{1}} (the case p 0 = p 1 {\textstyle p_{0}=p_{1}} trivially follows from ( 1 )). Let f {\textstyle f} be
44583-583: The limit for n → ∞ {\textstyle n\to \infty } . Since | f n | ≤ | f | {\textstyle \left\vert f_{n}\right\vert \leq \left\vert f\right\vert } , | g n | ≤ | g | {\textstyle \left\vert g_{n}\right\vert \leq \left\vert g\right\vert } and | h n | ≤ | h | {\textstyle \left\vert h_{n}\right\vert \leq \left\vert h\right\vert } , by
44814-419: The line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm: ‖ x ‖ 2 ≤ ‖ x ‖ 1 . {\displaystyle \|x\|_{2}\leq \|x\|_{1}.} This fact generalizes to p {\displaystyle p} -norms in that
45045-854: The linear functional f {\displaystyle f} and seminorm p {\displaystyle p} is encountered in many versions of the Hahn–Banach theorem. The Cartesian product X × Y {\displaystyle X\times Y} of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as ‖ ( x , y ) ‖ 1 = ‖ x ‖ + ‖ y ‖ , ‖ ( x , y ) ‖ ∞ = max ( ‖ x ‖ , ‖ y ‖ ) {\displaystyle \|(x,y)\|_{1}=\|x\|+\|y\|,\qquad \|(x,y)\|_{\infty }=\max(\|x\|,\|y\|)} which correspond (respectively) to
45276-465: The linearity of T {\textstyle T} as an operator of types ( p 0 , q 0 ) {\textstyle (p_{0},q_{0})} and ( p 1 , q 1 ) {\textstyle (p_{1},q_{1})} (we have not proven yet that it is of type ( p θ , q θ ) {\textstyle (p_{\theta },q_{\theta })} for
45507-573: The norm by using the polarization identity . On ℓ 2 , {\displaystyle \ell ^{2},} it can be defined by ⟨ ( x i ) i , ( y n ) i ⟩ ℓ 2 = ∑ i x i y i ¯ {\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}} while for
45738-427: The norm topology τ {\displaystyle \tau } on X , {\displaystyle X,} then ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} is a Banach space. A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric. Every Banach space
45969-482: The origin if and only if f {\displaystyle f} is uniformly continuous on all of X {\displaystyle X} ; and if in addition f ( 0 ) = 0 {\displaystyle f(0)=0} then f {\displaystyle f} is continuous if and only if its absolute value | f | : X → [ 0 , ∞ ) {\displaystyle |f|:X\to [0,\infty )}
46200-426: The origin in this metric is "concave", the topology defined on R n {\displaystyle \mathbb {R} ^{n}} by the metric B p {\displaystyle B_{p}} is the usual vector space topology of R n , {\displaystyle \mathbb {R} ^{n},} hence ℓ n p {\displaystyle \ell _{n}^{p}}
46431-459: The other hand, if f {\displaystyle f} is a measurable function for which there exists some 0 < p ≤ ∞ {\displaystyle 0<p\leq \infty } such that ‖ f ‖ p = 0 {\displaystyle \|f\|_{p}=0} then f = 0 {\displaystyle f=0} almost everywhere. When p {\displaystyle p}
46662-498: The other hand, if we take the layer-cake decomposition f = f 1 {| f |>1} + f 1 {| f |≤1} , then we see that f 1 {| f |>1} ∈ L and f 1 {| f |≤1} ∈ L , whence we obtain the following result: Proposition — Each f in L can be written as
46893-504: The product | f | = | f | | f | and applying Hölder's inequality to its p θ power, we obtain the following result, foundational in the study of L -spaces: Proposition (log-convexity of L -norms) — Each f ∈ L ∩ L satisfies: This result, whose name derives from the convexity of the map 1 ⁄ p ↦ log || f || p on [0, ∞] , implies that L ∩ L ⊂ L . On
47124-457: The quotient is the null space. The closed linear subspace M {\displaystyle M} of X {\displaystyle X} is said to be a complemented subspace of X {\displaystyle X} if M {\displaystyle M} is the range of a surjective bounded linear projection P : X → M . {\displaystyle P:X\to M.} In this case,
47355-401: The resulting function does not define a norm, because it is not subadditive . On the other hand, the formula | x 1 | p + | x 2 | p + ⋯ + | x n | p {\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}} defines a subadditive function at
47586-585: The right means that only countably many summands are nonzero (see also Unconditional convergence ). With the norm ‖ x ‖ p = ( ∑ i ∈ I | x i | p ) 1 / p {\displaystyle \|x\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}} the space ℓ p ( I ) {\displaystyle \ell ^{p}(I)} becomes
47817-629: The role of the operator and the operand, or in other words, we think of S as the operator of convolution with g , and get that S is bounded from L to L . Further, since g is in L we get, in view of Hölder's inequality, that S is bounded from L to L , where again 1 / p + 1 / q = 1 . So interpolating we get ‖ f ∗ g ‖ s ≤ ‖ f ‖ r ‖ g ‖ p {\displaystyle \|f*g\|_{s}\leq \|f\|_{r}\|g\|_{p}} where
48048-673: The same zero set (it does not depend on p {\displaystyle p} ). So denote this common set by N = def { f : f = 0 μ -almost everywhere } = { f ∈ L p ( S , μ ) : ‖ f ‖ p = 0 } ∀ p . {\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.} This set
48279-407: The same . Therefore, the problem of studying operators on the sumset L + L essentially reduces to the study of operators that map two natural domain spaces, L and L , boundedly to two target spaces: L and L , respectively. Since such operators map the sumset space L + L to L + L , it is natural to expect that these operators map the intermediate space L to
48510-1630: The same process for z = 1 + i y {\textstyle z=1+\mathrm {i} y} to obtain | f 1 + i y ( ξ ) | = | f ( ξ ) | p θ / p 1 {\textstyle \left\vert f_{1+\mathrm {i} y}(\xi )\right\vert =\left\vert f(\xi )\right\vert ^{p_{\theta }/p_{1}}} , | g 1 + i y ( ζ ) | = | g ( ζ ) | q θ ′ / q 1 ′ {\textstyle \left\vert g_{1+\mathrm {i} y}(\zeta )\right\vert =\left\vert g(\zeta )\right\vert ^{q_{\theta }'/q_{1}'}} and, finally, | Φ ( 1 + i y ) | ≤ ‖ T ‖ L p 1 → L q 1 ‖ f 1 + i y ‖ p 1 ‖ g 1 + i y ‖ q 1 ′ = ‖ T ‖ L p 1 → L q 1 . {\displaystyle \left\vert \Phi (1+\mathrm {i} y)\right\vert \leq \|T\|_{L^{p_{1}}\to L^{q_{1}}}\lVert f_{1+\mathrm {i} y}\rVert _{p_{1}}\lVert g_{1+\mathrm {i} y}\rVert _{q_{1}'}=\|T\|_{L^{p_{1}}\to L^{q_{1}}}.} So far, we have proven that when f {\textstyle f}
48741-414: The scalar field (which is R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional . This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness
48972-531: The scalar field to be C . For extended-real-valued functions, this restriction can be bypassed by redefining the function to be finite everywhere—possible, as every integrable function must be finite almost everywhere. A more serious disadvantage is that, in practice, many operators, such as the Hardy–Littlewood maximal operator and the Calderón–Zygmund operators , do not have good endpoint estimates. In
49203-767: The scalar multiple C B n p {\displaystyle C\,B_{n}^{p}} of the p {\displaystyle p} -unit ball contains the convex hull of B n p , {\displaystyle B_{n}^{p},} which is equal to B n 1 . {\displaystyle B_{n}^{1}.} The fact that for fixed p < 1 {\displaystyle p<1} we have C p ( n ) = n 1 p − 1 → ∞ , as n → ∞ {\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty } shows that
49434-423: The separable Hilbert sequence space ℓ 2 ( N ) {\displaystyle \ \ell ^{2}(\mathbb {N} )\ } with its usual norm ‖ ⋅ ‖ 2 . {\displaystyle \ \|\cdot \|_{2}~.} The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space
49665-424: The set L p ( S , μ ) {\displaystyle {\mathcal {L}}^{p}(S,\,\mu )} of all measurable functions f {\displaystyle f} from S {\displaystyle S} to C {\displaystyle \mathbb {C} } or R {\displaystyle \mathbb {R} } whose absolute value raised to
49896-606: The set for p = ∞ , {\displaystyle p=\infty ,} recall that two functions f {\displaystyle f} and g {\displaystyle g} defined on S {\displaystyle S} are said to be equal almost everywhere , written f = g {\displaystyle f=g} a.e. , if the set { s ∈ S : f ( s ) ≠ g ( s ) } {\displaystyle \{s\in S:f(s)\neq g(s)\}}
50127-449: The space ℓ 2 {\displaystyle \ell ^{2}} of square summable sequences; the space c 0 {\displaystyle c_{0}} of sequences tending to zero and the space ℓ ∞ {\displaystyle \ell ^{\infty }} of bounded sequences; the space C ( K ) {\displaystyle C(K)} of continuous scalar functions on
50358-486: The space L 2 {\displaystyle L^{2}} is a Hilbert space. The Hardy spaces , the Sobolev spaces are examples of Banach spaces that are related to L p {\displaystyle L^{p}} spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others. A Banach algebra
50589-633: The space L 2 ( X , μ ) {\displaystyle L^{2}(X,\mu )} associated with a measure space ( X , Σ , μ ) , {\displaystyle (X,\Sigma ,\mu ),} which consists of all square-integrable functions , it is ⟨ f , g ⟩ L 2 = ∫ X f ( x ) g ( x ) ¯ d x . {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}f(x){\overline {g(x)}}\,\mathrm {d} x.} Now consider
50820-603: The space X {\displaystyle X} is isomorphic to the direct sum of M {\displaystyle M} and ker P , {\displaystyle \ker P,} the kernel of the projection P . {\displaystyle P.} Suppose that X {\displaystyle X} and Y {\displaystyle Y} are Banach spaces and that T ∈ B ( X , Y ) . {\displaystyle T\in B(X,Y).} There exists
51051-485: The space of all μ 2 -measurable functions on Ω 2 . We assume the following further properties on this collection of linear operators: Then, for each 0 < θ < 1 , the operator T θ maps L ( μ 1 ) boundedly into L ( μ 2 ) . The theory of real Hardy spaces and the space of bounded mean oscillations permits us to wield the Stein interpolation theorem argument in dealing with operators on
51282-839: The squared L 2 {\displaystyle L^{2}} norm of the parameter vector. The Fourier transform for the real line (or, for periodic functions , see Fourier series ), maps L p ( R ) {\displaystyle L^{p}(\mathbb {R} )} to L q ( R ) {\displaystyle L^{q}(\mathbb {R} )} (or L p ( T ) {\displaystyle L^{p}(\mathbf {T} )} to ℓ q {\displaystyle \ell ^{q}} ) respectively, where 1 ≤ p ≤ 2 {\displaystyle 1\leq p\leq 2} and 1 p + 1 q = 1. {\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.} This
51513-1166: The supremum is taken over all simple functions g {\textstyle g} in L p ′ ( μ 1 ) {\textstyle L^{p'}(\mu _{1})} such that ‖ g ‖ p ′ ≤ 1 {\textstyle \lVert g\rVert _{p'}\leq 1} . In our case, the lemma above implies ‖ T f ‖ q θ ≤ ‖ T ‖ L p 0 → L q 0 1 − θ ‖ T ‖ L p 1 → L q 1 θ {\displaystyle \lVert Tf\rVert _{q_{\theta }}\leq \|T\|_{L^{p_{0}}\to L^{q_{0}}}^{1-\theta }\|T\|_{L^{p_{1}}\to L^{q_{1}}}^{\theta }} for all simple function f {\textstyle f} with ‖ f ‖ p θ = 1 {\textstyle \lVert f\rVert _{p_{\theta }}=1} . Equivalently, for
51744-582: The supremum is taken with respect to all g {\textstyle g} simple functions with ‖ g ‖ q θ ′ = 1 {\textstyle \lVert g\rVert _{q_{\theta }'}=1} . The left-hand side can be rewritten by means of the following lemma. Lemma — Let 1 ≤ p , p ′ ≤ ∞ {\textstyle 1\leq p,p'\leq \infty } be conjugate exponents and let f {\textstyle f} be
51975-1248: The theorem statement. Next, we define the two complex functions u : C → C v : C → C z ↦ u ( z ) = 1 − z p 0 + z p 1 z ↦ v ( z ) = 1 − z q 0 + z q 1 . {\displaystyle {\begin{aligned}u:\mathbb {C} &\to \mathbb {C} &v:\mathbb {C} &\to \mathbb {C} \\z&\mapsto u(z)={\frac {1-z}{p_{0}}}+{\frac {z}{p_{1}}}&z&\mapsto v(z)={\frac {1-z}{q_{0}}}+{\frac {z}{q_{1}}}.\end{aligned}}} Note that, for z = θ {\textstyle z=\theta } , u ( θ ) = p θ − 1 {\textstyle u(\theta )=p_{\theta }^{-1}} and v ( θ ) = q θ − 1 {\textstyle v(\theta )=q_{\theta }^{-1}} . We then extend f {\textstyle f} and g {\textstyle g} to depend on
52206-414: The theory of uniform spaces . Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity , called the canonical uniformity , that depends only on vector subtraction and the topology τ {\displaystyle \tau } that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced
52437-399: The topology τ {\displaystyle \tau } (and even applies to TVSs that are not even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space. If ( X , τ ) {\displaystyle (X,\tau )}
52668-459: The topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms , which are generalizations of norms . It is even possible for a Fréchet space to have a topology that is induced by a countable family of norms (such norms would necessarily be continuous) but to not be a Banach/ normable space because its topology can not be defined by any single norm. An example of such
52899-626: The value under ‖ ⋅ ‖ p {\displaystyle \|\,\cdot \,\|_{p}} of a measurable function f {\displaystyle f} and its absolute value | f | : S → [ 0 , ∞ ] {\displaystyle |f|:S\to [0,\infty ]} are always the same (that is, ‖ f ‖ p = ‖ | f | ‖ p {\displaystyle \|f\|_{p}=\||f|\|_{p}} for all p {\displaystyle p} ) and so
53130-502: The vector space B ( X , Y ) {\displaystyle B(X,Y)} can be given the operator norm ‖ T ‖ = sup { ‖ T x ‖ Y ∣ x ∈ X , ‖ x ‖ X ≤ 1 } . {\displaystyle \|T\|=\sup \left\{\|Tx\|_{Y}\mid x\in X,\ \|x\|_{X}\leq 1\right\}.} For Y {\displaystyle Y}
53361-577: Was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space ". Banach spaces originally grew out of the study of function spaces by Hilbert , Fréchet , and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis , the spaces under study are often Banach spaces. A Banach space is a complete normed space ( X , ‖ ⋅ ‖ ) . {\displaystyle \ (X,\|\cdot \|)~.} A normed space
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