Misplaced Pages

#605394

100-400: A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space . In a similar manner, a vector space with a seminorm is called a seminormed vector space . The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy

200-503: A + b i + c j + d k {\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} } in H . {\displaystyle \mathbb {H} .} This is the same as the Euclidean norm on H {\displaystyle \mathbb {H} } considered as the vector space R 4 . {\displaystyle \mathbb {R} ^{4}.} Similarly,

300-409: A Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis . Hence, the Euclidean norm can be written in a coordinate-free way as ‖ x ‖ := x ⋅ x . {\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} The Euclidean norm is also called

400-539: A sublinear function . A map p : X → R {\displaystyle p:X\to \mathbb {R} } is called a sublinear function if it is subadditive and positive homogeneous . Unlike a seminorm, a sublinear function is not necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem . A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} }

500-510: A vector space quotient X / W , {\displaystyle X/W,} where W {\displaystyle W} is the subspace of X {\displaystyle X} consisting of all vectors x ∈ X {\displaystyle x\in X} with p ( x ) = 0. {\displaystyle p(x)=0.} Then X / W {\displaystyle X/W} carries

600-435: A complete metric topological vector space . These spaces are of great interest in functional analysis , probability theory and harmonic analysis . However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional. The partial derivative of the p {\displaystyle p} -norm

700-438: A field A , {\displaystyle A,} an involution ∗ , {\displaystyle \,*,} and a quadratic form N , {\displaystyle N,} which is called the "norm". In several cases N {\displaystyle N} is an isotropic quadratic form so that A {\displaystyle A} has at least one null vector , contrary to

800-462: A non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous. In signal processing and statistics , David Donoho referred to the zero " norm " with quotation marks. Following Donoho's notation, the zero "norm" of x {\displaystyle x} is simply the number of non-zero coordinates of x , {\displaystyle x,} or

900-468: A norm p : X → R {\displaystyle p:X\to \mathbb {R} } is given on a vector space X , {\displaystyle X,} then the norm of a vector z ∈ X {\displaystyle z\in X} is usually denoted by enclosing it within double vertical lines: ‖ z ‖ = p ( z ) . {\displaystyle \|z\|=p(z).} Such notation

1000-468: A norm defined by p ( x + W ) = p ( x ) . {\displaystyle p(x+W)=p(x).} The resulting topology, pulled back to X , {\displaystyle X,} is precisely the topology induced by p . {\displaystyle p.} Any seminorm-induced topology makes X {\displaystyle X} locally convex , as follows. If p {\displaystyle p}

1100-486: A scalar s {\displaystyle s} : A seminorm on X {\displaystyle X} is a function p : X → R {\displaystyle p:X\to \mathbb {R} } that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional ). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if p {\displaystyle p}

SECTION 10

#1732787737606

1200-419: A seminorm p {\displaystyle p} on X . {\displaystyle X.} If the seminorm p {\displaystyle p} is also a norm then the seminormed space ( X , p ) {\displaystyle (X,p)} is called a normed space . Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called

1300-860: A synonym of "non-negative"; these definitions are not equivalent. Suppose that p {\displaystyle p} and q {\displaystyle q} are two norms (or seminorms) on a vector space X . {\displaystyle X.} Then p {\displaystyle p} and q {\displaystyle q} are called equivalent , if there exist two positive real constants c {\displaystyle c} and C {\displaystyle C} such that for every vector x ∈ X , {\displaystyle x\in X,} c q ( x ) ≤ p ( x ) ≤ C q ( x ) . {\displaystyle cq(x)\leq p(x)\leq Cq(x).} The relation " p {\displaystyle p}

1400-466: A topology, called the seminorm-induced topology , via the canonical translation-invariant pseudometric d p : X × X → R {\displaystyle d_{p}:X\times X\to \mathbb {R} } ; d p ( x , y ) := p ( x − y ) = p ( y − x ) . {\displaystyle d_{p}(x,y):=p(x-y)=p(y-x).} This topology

1500-528: A two-dimensional vector space over the reals; the absolute value is a norm for these two structures. Any norm p {\displaystyle p} on a one-dimensional vector space X {\displaystyle X} is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces f : F → X , {\displaystyle f:\mathbb {F} \to X,} where F {\displaystyle \mathbb {F} }

1600-539: A vector in the Euclidean plane, makes the quantity x 2 + y 2 {\textstyle {\sqrt {x^{2}+y^{2}}}} (as first suggested by Euler) the Euclidean norm associated with the complex number. For z = x + i y {\displaystyle z=x+iy} , the norm can also be written as z ¯ z {\displaystyle {\sqrt {{\bar {z}}z}}} where z ¯ {\displaystyle {\bar {z}}}

1700-592: A vector of norm 1 , {\displaystyle 1,} which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm. On the n {\displaystyle n} -dimensional Euclidean space R n , {\displaystyle \mathbb {R} ^{n},} the intuitive notion of length of the vector x = ( x 1 , x 2 , … , x n ) {\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)}

1800-628: Is ∂ ‖ x ‖ p ∂ x = x ∘ | x | p − 2 ‖ x ‖ p p − 1 . {\displaystyle {\frac {\partial \|\mathbf {x} \|_{p}}{\partial \mathbf {x} }}={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.} where ∘ {\displaystyle \circ } denotes Hadamard product and | ⋅ | {\displaystyle |\cdot |}

1900-478: Is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}.} For p = 1 , {\displaystyle p=1,} we get the taxicab norm , for p = 2 {\displaystyle p=2} we get

2000-474: Is stronger than p {\displaystyle p} and that p {\displaystyle p} is weaker than q {\displaystyle q} if any of the following equivalent conditions holds: The seminorms p {\displaystyle p} and q {\displaystyle q} are called equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of

2100-408: Is Hausdorff if and only if d p {\displaystyle d_{p}} is a metric, which occurs if and only if p {\displaystyle p} is a norm . This topology makes X {\displaystyle X} into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at

SECTION 20

#1732787737606

2200-738: Is a Hamel basis for a vector space X {\displaystyle X} then the real-valued map that sends x = ∑ i ∈ I s i x i ∈ X {\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X} (where all but finitely many of the scalars s i {\displaystyle s_{i}} are 0 {\displaystyle 0} ) to ∑ i ∈ I | s i | {\displaystyle \sum _{i\in I}\left|s_{i}\right|}

2300-392: Is a T 1 space and admits a bounded convex neighborhood of the origin. If X {\displaystyle X} is a Hausdorff locally convex TVS then the following are equivalent: Furthermore, X {\displaystyle X} is finite dimensional if and only if X σ ′ {\displaystyle X_{\sigma }^{\prime }}

2400-510: Is a balanced function . A seminorm p {\displaystyle p} is a norm on X {\displaystyle X} if and only if { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in X:p(x)<;1\}} does not contain a non-trivial vector subspace. If p : X → [ 0 , ∞ ) {\displaystyle p:X\to [0,\infty )}

2500-403: Is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable. Let p : X → R {\displaystyle p:X\to \mathbb {R} } be a non-negative function. The following are equivalent: If any of the above conditions hold, then the following are equivalent: If p {\displaystyle p}

2600-469: Is a given constant, c , {\displaystyle c,} forms the surface of a hypercube with edge length 2 c . {\displaystyle 2c.} The energy norm of a vector x = ( x 1 , x 2 , … , x n ) ∈ R n {\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}

2700-508: Is a given positive constant forms an n {\displaystyle n} -sphere . The Euclidean norm of a complex number is the absolute value (also called the modulus ) of it, if the complex plane is identified with the Euclidean plane R 2 . {\displaystyle \mathbb {R} ^{2}.} This identification of the complex number x + i y {\displaystyle x+iy} as

2800-444: Is a linear functional on X {\displaystyle X} then: Seminorms offer a particularly clean formulation of the Hahn–Banach theorem : A similar extension property also holds for seminorms: Theorem   (Extending seminorms)  —  If M {\displaystyle M} is a vector subspace of X , {\displaystyle X,} p {\displaystyle p}

2900-616: Is a linear map between seminormed spaces then the following are equivalent: If F {\displaystyle F} is continuous then q ( F ( x ) ) ≤ ‖ F ‖ p , q p ( x ) {\displaystyle q(F(x))\leq \|F\|_{p,q}p(x)} for all x ∈ X . {\displaystyle x\in X.} The space of all continuous linear maps F : ( X , p ) → ( Y , q ) {\displaystyle F:(X,p)\to (Y,q)} between seminormed spaces

3000-451: Is a norm (or more generally, a seminorm) then p ( 0 ) = 0 {\displaystyle p(0)=0} and that p {\displaystyle p} also has the following property: Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined " positive " to be a synonym of "positive definite", some authors instead define " positive " to be

3100-536: Is a norm on R 4 . {\displaystyle \mathbb {R} ^{4}.} Seminorm In mathematics , particularly in functional analysis , a seminorm is a norm that need not be positive definite . Seminorms are intimately connected with convex sets : every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set

Norm (mathematics) - Misplaced Pages Continue

3200-424: Is a norm on X . {\displaystyle X.} There are also a large number of norms that exhibit additional properties that make them useful for specific problems. The absolute value | x | {\displaystyle |x|} is a norm on the vector space formed by the real or complex numbers . The complex numbers form a one-dimensional vector space over themselves and

3300-488: Is a normed space and x , y ∈ X {\displaystyle x,y\in X} then ‖ x − y ‖ = ‖ x − z ‖ + ‖ z − y ‖ {\displaystyle \|x-y\|=\|x-z\|+\|z-y\|} for all z {\displaystyle z} in the interval [ x , y ] . {\displaystyle [x,y].} Every norm

3400-419: Is a seminorm if and only if it is a sublinear and balanced function . Seminorms on a vector space X {\displaystyle X} are intimately tied, via Minkowski functionals, to subsets of X {\displaystyle X} that are convex , balanced , and absorbing . Given such a subset D {\displaystyle D} of X , {\displaystyle X,}

3500-741: Is a seminorm on M , {\displaystyle M,} and q {\displaystyle q} is a seminorm on X {\displaystyle X} such that p ≤ q | M , {\displaystyle p\leq q{\big \vert }_{M},} then there exists a seminorm P {\displaystyle P} on X {\displaystyle X} such that P | M = p {\displaystyle P{\big \vert }_{M}=p} and P ≤ q . {\displaystyle P\leq q.} A seminorm p {\displaystyle p} on X {\displaystyle X} induces

3600-401: Is a seminorm on X {\displaystyle X} and r ∈ R , {\displaystyle r\in \mathbb {R} ,} call the set { x ∈ X : p ( x ) < r } {\displaystyle \{x\in X:p(x)<r\}} the open ball of radius r {\displaystyle r} about the origin ; likewise

3700-398: Is a seminorm on X {\displaystyle X} then ker ⁡ p := p − 1 ( 0 ) {\displaystyle \ker p:=p^{-1}(0)} is a vector subspace of X {\displaystyle X} and for every x ∈ X , {\displaystyle x\in X,} p {\displaystyle p}

3800-422: Is a seminorm on a topological vector space X , {\displaystyle X,} then the following are equivalent: In particular, if ( X , p ) {\displaystyle (X,p)} is a seminormed space then a seminorm q {\displaystyle q} on X {\displaystyle X} is continuous if and only if q {\displaystyle q}

3900-496: Is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Let X {\displaystyle X} be a vector space over either the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C . {\displaystyle \mathbb {C} .} A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} }

4000-629: Is a set satisfying { x ∈ X : p ( x ) < 1 } ⊆ D ⊆ { x ∈ X : p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)<1\}\subseteq D\subseteq \{x\in X:p(x)\leq 1\}} then D {\displaystyle D} is absorbing in X {\displaystyle X} and p = p D {\displaystyle p=p_{D}} where p D {\displaystyle p_{D}} denotes

4100-490: Is a sublinear function on a real vector space X {\displaystyle X} then the following are equivalent: If p , q : X → [ 0 , ∞ ) {\displaystyle p,q:X\to [0,\infty )} are seminorms on X {\displaystyle X} then: If p {\displaystyle p} is a seminorm on X {\displaystyle X} and f {\displaystyle f}

Norm (mathematics) - Misplaced Pages Continue

4200-411: Is a topological vector space that possesses a bounded neighborhood of the origin. Normability of topological vector spaces is characterized by Kolmogorov's normability criterion . A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin. Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set. A TVS is normable if and only if it

4300-457: Is a vector space, and it is also true that the function ∫ X | f ( x ) − g ( x ) | p   d μ {\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu } (without p {\displaystyle p} th root) defines a distance that makes L p ( X ) {\displaystyle L^{p}(X)} into

4400-589: Is also sometimes used if p {\displaystyle p} is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below ), the notation | x | {\displaystyle |x|} with single vertical lines is also widespread. Every (real or complex) vector space admits a norm: If x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}

4500-1009: Is an absorbing disk in X . {\displaystyle X.} If p {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} then there exists a linear functional f {\displaystyle f} on X {\displaystyle X} such that f ≤ p {\displaystyle f\leq p} and furthermore, for any linear functional g {\displaystyle g} on X , {\displaystyle X,} g ≤ p {\displaystyle g\leq p} on X {\displaystyle X} if and only if g − 1 ( 1 ) ∩ { x ∈ X : p ( x ) < 1 } = ∅ . {\displaystyle g^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .} Other properties of seminorms Every seminorm

4600-443: Is an acronym for the s quare r oot of the s um of s quares. The Euclidean norm is by far the most commonly used norm on R n , {\displaystyle \mathbb {R} ^{n},} but there are other norms on this vector space as will be shown below. However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces. The inner product of two vectors of

4700-521: Is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some b ≤ 1 {\displaystyle b\leq 1} such that p ( x + y ) ≤ b p ( p ( x ) + p ( y ) )  for all  x , y ∈ X . {\displaystyle p(x+y)\leq bp(p(x)+p(y)){\text{ for all }}x,y\in X.} The smallest value of b {\displaystyle b} for which this holds

4800-409: Is called a seminorm if it satisfies the following two conditions: These two conditions imply that p ( 0 ) = 0 {\displaystyle p(0)=0} and that every seminorm p {\displaystyle p} also has the following property: Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this

4900-534: Is captured by the formula ‖ x ‖ 2 := x 1 2 + ⋯ + x n 2 . {\displaystyle \|{\boldsymbol {x}}\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.} This is the Euclidean norm , which gives the ordinary distance from the origin to the point X —a consequence of the Pythagorean theorem . This operation may also be referred to as "SRSS", which

5000-751: Is clear that if A {\displaystyle A} is the identity matrix , this norm corresponds to the Euclidean norm . If A {\displaystyle A} is diagonal, this norm is also called a weighted norm . The energy norm is induced by the inner product given by ⟨ x , y ⟩ A := x T ⋅ A ⋅ x {\displaystyle \langle {\boldsymbol {x}},{\boldsymbol {y}}\rangle _{A}:={\boldsymbol {x}}^{T}\cdot A\cdot {\boldsymbol {x}}} for x , y ∈ R n {\displaystyle {\boldsymbol {x}},{\boldsymbol {y}}\in \mathbb {R} ^{n}} . In general,

5100-862: Is constant on the set x + ker ⁡ p = { x + k : p ( k ) = 0 } {\displaystyle x+\ker p=\{x+k:p(k)=0\}} and equal to p ( x ) . {\displaystyle p(x).} Furthermore, for any real r > 0 , {\displaystyle r>0,} r { x ∈ X : p ( x ) < 1 } = { x ∈ X : p ( x ) < r } = { x ∈ X : 1 r p ( x ) < 1 } . {\displaystyle r\{x\in X:p(x)<1\}=\{x\in X:p(x)<r\}=\left\{x\in X:{\tfrac {1}{r}}p(x)<1\right\}.} If D {\displaystyle D}

SECTION 50

#1732787737606

5200-641: Is continuous. If F : ( X , p ) → ( Y , q ) {\displaystyle F:(X,p)\to (Y,q)} is a map between seminormed spaces then let ‖ F ‖ p , q := sup { q ( F ( x ) ) : p ( x ) ≤ 1 , x ∈ X } . {\displaystyle \|F\|_{p,q}:=\sup\{q(F(x)):p(x)\leq 1,x\in X\}.} If F : ( X , p ) → ( Y , q ) {\displaystyle F:(X,p)\to (Y,q)}

5300-438: Is defined by ‖ q ‖ = q q ∗   = q ∗ q   = a 2 + b 2 + c 2 + d 2   {\displaystyle \lVert q\rVert ={\sqrt {\,qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}} for every quaternion q =

5400-438: Is defined in terms of a symmetric positive definite matrix A ∈ R n {\displaystyle A\in \mathbb {R} ^{n}} as ‖ x ‖ A := x T ⋅ A ⋅ x . {\displaystyle {\|{\boldsymbol {x}}\|}_{A}:={\sqrt {{\boldsymbol {x}}^{T}\cdot A\cdot {\boldsymbol {x}}}}.} It

5500-635: Is dominated by a positive scalar multiple of p . {\displaystyle p.} If X {\displaystyle X} is a real TVS, f {\displaystyle f} is a linear functional on X , {\displaystyle X,} and p {\displaystyle p} is a continuous seminorm (or more generally, a sublinear function) on X , {\displaystyle X,} then f ≤ p {\displaystyle f\leq p} on X {\displaystyle X} implies that f {\displaystyle f}

5600-415: Is either R {\displaystyle \mathbb {R} } or C , {\displaystyle \mathbb {C} ,} and norm-preserving means that | x | = p ( f ( x ) ) . {\displaystyle |x|=p(f(x)).} This isomorphism is given by sending 1 ∈ F {\displaystyle 1\in \mathbb {F} } to

5700-439: Is equivalent to q {\displaystyle q} " is reflexive , symmetric ( c q ≤ p ≤ C q {\displaystyle cq\leq p\leq Cq} implies 1 C p ≤ q ≤ 1 c p {\displaystyle {\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p} ), and transitive and thus defines an equivalence relation on

5800-556: Is given by ∂ ∂ x k ‖ x ‖ p = x k | x k | p − 2 ‖ x ‖ p p − 1 . {\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {x_{k}\left|x_{k}\right|^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.} The derivative with respect to x , {\displaystyle x,} therefore,

5900-490: Is itself a seminormed space under the seminorm ‖ F ‖ p , q . {\displaystyle \|F\|_{p,q}.} This seminorm is a norm if q {\displaystyle q} is a norm. The concept of norm in composition algebras does not share the usual properties of a norm. A composition algebra ( A , ∗ , N ) {\displaystyle (A,*,N)} consists of an algebra over

6000-448: Is normable (here X σ ′ {\displaystyle X_{\sigma }^{\prime }} denotes X ′ {\displaystyle X^{\prime }} endowed with the weak-* topology ). The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional). If p {\displaystyle p}

6100-454: Is not a norm because it may yield negative results. Let p ≥ 1 {\displaystyle p\geq 1} be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell ^{p}} -norm) of vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}

SECTION 60

#1732787737606

6200-461: Is not a norm in the usual sense because it lacks the required homogeneity property. In metric geometry , the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance , which is important in coding and information theory . In the field of real or complex numbers,

6300-425: Is not necessary since it follows from the other two properties. By definition, a norm on X {\displaystyle X} is a seminorm that also separates points, meaning that it has the following additional property: A seminormed space is a pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and

6400-794: Is related to the generalized mean or power mean. For p = 2 , {\displaystyle p=2,} the ‖ ⋅ ‖ 2 {\displaystyle \|\,\cdot \,\|_{2}} -norm is even induced by a canonical inner product ⟨ ⋅ , ⋅ ⟩ , {\displaystyle \langle \,\cdot ,\,\cdot \rangle ,} meaning that ‖ x ‖ 2 = ⟨ x , x ⟩ {\textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all vectors x . {\displaystyle \mathbf {x} .} This inner product can be expressed in terms of

6500-573: Is some vector such that x = ( x 1 , x 2 , … , x n ) , {\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),} then: ‖ x ‖ ∞ := max ( | x 1 | , … , | x n | ) . {\displaystyle \|\mathbf {x} \|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).} The set of vectors whose infinity norm

6600-459: Is still of some interest for 0 < p < 1 , {\displaystyle 0<p<1,} but the resulting function does not define a norm, because it violates the triangle inequality . What is true for this case of 0 < p < 1 , {\displaystyle 0<p<1,} even in the measurable analog, is that the corresponding L p {\displaystyle L^{p}} class

6700-441: Is the complex conjugate of z . {\displaystyle z\,.} There are exactly four Euclidean Hurwitz algebras over the real numbers . These are the real numbers R , {\displaystyle \mathbb {R} ,} the complex numbers C , {\displaystyle \mathbb {C} ,} the quaternions H , {\displaystyle \mathbb {H} ,} and lastly

6800-869: Is used for absolute value of each component of the vector. For the special case of p = 2 , {\displaystyle p=2,} this becomes ∂ ∂ x k ‖ x ‖ 2 = x k ‖ x ‖ 2 , {\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{2}={\frac {x_{k}}{\|\mathbf {x} \|_{2}}},} or ∂ ∂ x ‖ x ‖ 2 = x ‖ x ‖ 2 . {\displaystyle {\frac {\partial }{\partial \mathbf {x} }}\|\mathbf {x} \|_{2}={\frac {\mathbf {x} }{\|\mathbf {x} \|_{2}}}.} If x {\displaystyle \mathbf {x} }

6900-700: Is valid for any inner product space , including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product . Hence the formula in this case can also be written using the following notation: ‖ x ‖ := x ⋅ x . {\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} ‖ x ‖ 1 := ∑ i = 1 n | x i | . {\displaystyle \|{\boldsymbol {x}}\|_{1}:=\sum _{i=1}^{n}\left|x_{i}\right|.} The name relates to

7000-502: The ℓ 1 {\displaystyle \ell ^{1}} norm . The distance derived from this norm is called the Manhattan distance or ℓ 1 {\displaystyle \ell ^{1}} distance . The 1-norm is simply the sum of the absolute values of the columns. In contrast, ∑ i = 1 n x i {\displaystyle \sum _{i=1}^{n}x_{i}}

7100-440: The p {\displaystyle p} -topology on X . {\displaystyle X.} The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms . If p {\displaystyle p} and q {\displaystyle q} are seminorms on X , {\displaystyle X,} then we say that q {\displaystyle q}

7200-606: The Banach space article. Generally, these norms do not give the same topologies. For example, an infinite-dimensional ℓ p {\displaystyle \ell ^{p}} space gives a strictly finer topology than an infinite-dimensional ℓ q {\displaystyle \ell ^{q}} space when p < q . {\displaystyle p<q\,.} Other norms on R n {\displaystyle \mathbb {R} ^{n}} can be constructed by combining

7300-519: The Euclidean norm , and as p {\displaystyle p} approaches ∞ {\displaystyle \infty } the p {\displaystyle p} -norm approaches the infinity norm or maximum norm : ‖ x ‖ ∞ := max i | x i | . {\displaystyle \|\mathbf {x} \|_{\infty }:=\max _{i}\left|x_{i}\right|.} The p {\displaystyle p} -norm

7400-950: The Minkowski functional associated with D {\displaystyle D} (that is, the gauge of D {\displaystyle D} ). In particular, if D {\displaystyle D} is as above and q {\displaystyle q} is any seminorm on X , {\displaystyle X,} then q = p {\displaystyle q=p} if and only if { x ∈ X : q ( x ) < 1 } ⊆ D ⊆ { x ∈ X : q ( x ) ≤ } . {\displaystyle \{x\in X:q(x)<;1\}\subseteq D\subseteq \{x\in X:q(x)\leq \}.} If ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\,\cdot \,\|)}

7500-410: The complex numbers C , {\displaystyle \mathbb {C} ,} a norm on X {\displaystyle X} is a real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } with the following properties, where | s | {\displaystyle |s|} denotes the usual absolute value of

7600-597: The octonions O , {\displaystyle \mathbb {O} ,} where the dimensions of these spaces over the real numbers are 1 , 2 , 4 ,  and  8 , {\displaystyle 1,2,4,{\text{ and }}8,} respectively. The canonical norms on R {\displaystyle \mathbb {R} } and C {\displaystyle \mathbb {C} } are their absolute value functions, as discussed previously. The canonical norm on H {\displaystyle \mathbb {H} } of quaternions

7700-643: The quadratic norm , L 2 {\displaystyle L^{2}} norm , ℓ 2 {\displaystyle \ell ^{2}} norm , 2-norm , or square norm ; see L p {\displaystyle L^{p}} space . It defines a distance function called the Euclidean length , L 2 {\displaystyle L^{2}} distance , or ℓ 2 {\displaystyle \ell ^{2}} distance . The set of vectors in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} whose Euclidean norm

7800-1249: The reverse triangle inequality : | p ( x ) − p ( y ) | ≤ p ( x − y ) {\displaystyle |p(x)-p(y)|\leq p(x-y)} and also 0 ≤ max { p ( x ) , p ( − x ) } {\textstyle 0\leq \max\{p(x),p(-x)\}} and p ( x ) − p ( y ) ≤ p ( x − y ) . {\displaystyle p(x)-p(y)\leq p(x-y).} For any vector x ∈ X {\displaystyle x\in X} and positive real r > 0 : {\displaystyle r>0:} x + { y ∈ X : p ( y ) < r } = { y ∈ X : p ( x − y ) < r } {\displaystyle x+\{y\in X:p(y)<;r\}=\{y\in X:p(x-y)<;r\}} and furthermore, { x ∈ X : p ( x ) < r } {\displaystyle \{x\in X:p(x)<r\}}

7900-503: The Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of p {\displaystyle p} -norms as p {\displaystyle p} approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous . Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to

8000-552: The Minkowski functional of D {\displaystyle D} is a seminorm. Conversely, given a seminorm p {\displaystyle p} on X , {\displaystyle X,} the sets { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in X:p(x)<1\}} and { x ∈ X : p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)\leq 1\}} are convex, balanced, and absorbing and furthermore,

8100-458: The Minkowski functional of these two sets (as well as of any set lying "in between them") is p . {\displaystyle p.} Every seminorm is a sublinear function , and thus satisfies all properties of a sublinear function , including convexity , p ( 0 ) = 0 , {\displaystyle p(0)=0,} and for all vectors x , y ∈ X {\displaystyle x,y\in X} :

8200-408: The above; for example ‖ x ‖ := 2 | x 1 | + 3 | x 2 | 2 + max ( | x 3 | , 2 | x 4 | ) 2 {\displaystyle \|x\|:=2\left|x_{1}\right|+{\sqrt {3\left|x_{2}\right|^{2}+\max(\left|x_{3}\right|,2\left|x_{4}\right|)^{2}}}}

8300-655: The bounds are achieved if x {\displaystyle {\boldsymbol {x}}} coincides with the corresponding (normalized) eigenvectors. Based on the symmetric matrix square root A 1 / 2 {\displaystyle A^{1/2}} , the energy norm of a vector can be written in terms of the standard Euclidean norm as ‖ x ‖ A = ‖ A 1 / 2 x ‖ 2 . {\displaystyle {\|{\boldsymbol {x}}\|}_{A}={\|A^{1/2}{\boldsymbol {x}}\|}_{2}.} In probability and functional analysis,

8400-866: The canonical norm on the octonions is just the Euclidean norm on R 8 . {\displaystyle \mathbb {R} ^{8}.} On an n {\displaystyle n} -dimensional complex space C n , {\displaystyle \mathbb {C} ^{n},} the most common norm is ‖ z ‖ := | z 1 | 2 + ⋯ + | z n | 2 = z 1 z ¯ 1 + ⋯ + z n z ¯ n . {\displaystyle \|{\boldsymbol {z}}\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.} In this case,

8500-405: The closed ball of radius r {\displaystyle r} is { x ∈ X : p ( x ) ≤ r } . {\displaystyle \{x\in X:p(x)\leq r\}.} The set of all open (resp. closed) p {\displaystyle p} -balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in

8600-458: The distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan ) to get from the origin to the point x . {\displaystyle x.} The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope , which has dimension equal to the dimension of the vector space minus 1. The Taxicab norm is also called

8700-413: The distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like

8800-462: The following conditions: A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space ) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T 1 (because a TVS is Hausdorff if and only if it is a T 1 space ). A locally bounded topological vector space

8900-630: The limit as p → + ∞ {\displaystyle p\rightarrow +\infty } , giving a supremum norm , and are called ℓ ∞ {\displaystyle \ell ^{\infty }} and L ∞ . {\displaystyle L^{\infty }\,.} Any inner product induces in a natural way the norm ‖ x ‖ := ⟨ x , x ⟩ . {\textstyle \|x\|:={\sqrt {\langle x,x\rangle }}.} Other examples of infinite-dimensional normed vector spaces can be found in

9000-672: The norm by using the polarization identity . On ℓ 2 , {\displaystyle \ell ^{2},} this inner product is the Euclidean inner product defined by ⟨ ( x n ) n , ( y n ) n ⟩ ℓ 2   =   ∑ n x n ¯ y n {\displaystyle \langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}{\overline {x_{n}}}y_{n}} while for

9100-774: The norm can be expressed as the square root of the inner product of the vector and itself: ‖ x ‖ := x H   x , {\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},} where x {\displaystyle {\boldsymbol {x}}} is represented as a column vector [ x 1 x 2 … x n ] T {\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}} and x H {\displaystyle {\boldsymbol {x}}^{H}} denotes its conjugate transpose . This formula

9200-452: The origin consisting of the following open balls (or the closed balls) centered at the origin: { x ∈ X : p ( x ) < r }  or  { x ∈ X : p ( x ) ≤ r } {\displaystyle \{x\in X:p(x)<r\}\quad {\text{ or }}\quad \{x\in X:p(x)\leq r\}} as r > 0 {\displaystyle r>0} ranges over

9300-440: The positive reals. Every seminormed space ( X , p ) {\displaystyle (X,p)} should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called seminormable . Equivalently, every vector space X {\displaystyle X} with seminorm p {\displaystyle p} induces

9400-419: The same axioms as a norm, with the equality replaced by an inequality " ≤ {\displaystyle \,\leq \,} " in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set . Given a vector space X {\displaystyle X} over a subfield F {\displaystyle F} of

9500-1655: The scalar argument in scalar–vector multiplication and with respect to its vector argument. Abusing terminology , some engineers omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the L 0 {\displaystyle L^{0}} norm, echoing the notation for the Lebesgue space of measurable functions . The generalization of the above norms to an infinite number of components leads to ℓ p {\displaystyle \ell ^{p}} and L p {\displaystyle L^{p}} spaces for p ≥ 1 , {\displaystyle p\geq 1\,,} with norms ‖ x ‖ p = ( ∑ i ∈ N | x i | p ) 1 / p  and    ‖ f ‖ p , X = ( ∫ X | f ( x ) | p   d x ) 1 / p {\displaystyle \|x\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg )}^{1/p}{\text{ and }}\ \|f\|_{p,X}={\bigg (}\int _{X}|f(x)|^{p}~\mathrm {d} x{\bigg )}^{1/p}} for complex-valued sequences and functions on X ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} respectively, which can be further generalized (see Haar measure ). These norms are also valid in

9600-672: The separation of points required for the usual norm discussed in this article. An ultraseminorm or a non-Archimedean seminorm is a seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } that also satisfies p ( x + y ) ≤ max { p ( x ) , p ( y ) }  for all  x , y ∈ X . {\displaystyle p(x+y)\leq \max\{p(x),p(y)\}{\text{ for all }}x,y\in X.} Weakening subadditivity: Quasi-seminorms A map p : X → R {\displaystyle p:X\to \mathbb {R} }

9700-417: The set of all norms on X . {\displaystyle X.} The norms p {\displaystyle p} and q {\displaystyle q} are equivalent if and only if they induce the same topology on X . {\displaystyle X.} Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces. If

9800-651: 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 , this inner product is ⟨ f , g ⟩ L 2 = ∫ X f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)}}g(x)\,\mathrm {d} x.} This definition

9900-495: The value of the norm is dependent on the spectrum of A {\displaystyle A} : For a vector x {\displaystyle {\boldsymbol {x}}} with a Euclidean norm of one, the value of ‖ x ‖ A {\displaystyle {\|{\boldsymbol {x}}\|}_{A}} is bounded from below and above by the smallest and largest absolute eigenvalues of A {\displaystyle A} respectively, where

10000-849: The zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm ( x n ) ↦ ∑ n 2 − n x n / ( 1 + x n ) . {\textstyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.} Here we mean by F-norm some real-valued function ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } on an F-space with distance d , {\displaystyle d,} such that ‖ x ‖ = d ( x , 0 ) . {\displaystyle \lVert x\rVert =d(x,0).} The F -norm described above

#605394