Misplaced Pages

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111-534: TVS may refer to: Mathematics [ edit ] Topological vector space Television [ edit ] Television Sydney , TV channel in Sydney, Australia Television South , ITV franchise holder in the South of England between 1982 and 1992 TVS Television Network , US distributor of live programming (mostly sports), in the 1960s and 1970s TVS (Poland) ,

222-419: A topological monomorphism , is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding . A topological vector space isomorphism (abbreviated TVS isomorphism ), also called a topological vector isomorphism or an isomorphism in the category of TVSs , is a bijective linear homeomorphism . Equivalently, it

333-618: A linear extension of f {\displaystyle f} to X , {\displaystyle X,} if it exists, is a linear map F : X → Y {\displaystyle F:X\to Y} defined on X {\displaystyle X} that extends f {\displaystyle f} (meaning that F ( s ) = f ( s ) {\displaystyle F(s)=f(s)} for all s ∈ S {\displaystyle s\in S} ) and takes its values from

444-427: A = 0 (one constraint), and in that case the solution space is ( x , b ) or equivalently stated, (0, b ) + ( x , 0), (one degree of freedom). The kernel may be expressed as the subspace ( x , 0) < V : the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R , ( a , b ) ↦ ( a ) {\textstyle (a,b)\mapsto (a)} : given

555-402: A linear mapping , linear transformation , vector space homomorphism , or in some contexts linear function ) is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves the operations of vector addition and scalar multiplication . The same names and the same definition are also used for the more general case of modules over

666-716: A matrix . This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A {\displaystyle A} is a real m × n {\displaystyle m\times n} matrix, then f ( x ) = A x {\displaystyle f(\mathbf {x} )=A\mathbf {x} } describes a linear map R n → R m {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}} (see Euclidean space ). Let { v 1 , … , v n } {\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}} be

777-415: A neighborhood basis at the origin for a vector topology on X . {\displaystyle X.} In this case, this topology is denoted by τ S {\displaystyle \tau _{\mathbb {S} }} and it is called the topology generated by S . {\displaystyle \mathbb {S} .} If S {\displaystyle \mathbb {S} }

888-418: A ring ). The multiplicative identity element of this algebra is the identity map id : V → V {\textstyle \operatorname {id} :V\to V} . An endomorphism of V {\textstyle V} that is also an isomorphism is called an automorphism of V {\textstyle V} . The composition of two automorphisms is again an automorphism, and

999-716: A ring ; see Module homomorphism . If a linear map is a bijection then it is called a linear isomorphism . In the case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism . Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V {\displaystyle V} and W {\displaystyle W} are real vector spaces (not necessarily with V = W {\displaystyle V=W} ), or it can be used to emphasize that V {\displaystyle V}

1110-724: A vector subspace of a real or complex vector space X {\displaystyle X} has a linear extension to all of X . {\displaystyle X.} Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional f {\displaystyle f} is dominated by some given seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } (meaning that | f ( m ) | ≤ p ( m ) {\displaystyle |f(m)|\leq p(m)} holds for all m {\displaystyle m} in

1221-744: A basis for V {\displaystyle V} . Then every vector v ∈ V {\displaystyle \mathbf {v} \in V} is uniquely determined by the coefficients c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} in the field R {\displaystyle \mathbb {R} } : v = c 1 v 1 + ⋯ + c n v n . {\displaystyle \mathbf {v} =c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}.} If f : V → W {\textstyle f:V\to W}

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1332-488: A basis for W {\displaystyle W} . Then we can represent each vector f ( v j ) {\displaystyle f(\mathbf {v} _{j})} as f ( v j ) = a 1 j w 1 + ⋯ + a m j w m . {\displaystyle f\left(\mathbf {v} _{j}\right)=a_{1j}\mathbf {w} _{1}+\cdots +a_{mj}\mathbf {w} _{m}.} Thus,

1443-575: A bit; E {\displaystyle E} is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also, E {\displaystyle E} is bounded if and only if for every balanced neighborhood V {\displaystyle V} of the origin, there exists t {\displaystyle t} such that E ⊆ t V . {\displaystyle E\subseteq tV.} Moreover, when X {\displaystyle X}

1554-581: A collection of strings is said to be τ {\displaystyle \tau } fundamental . Conversely, if X {\displaystyle X} is a vector space and if S {\displaystyle \mathbb {S} } is a collection of strings in X {\displaystyle X} that is directed downward, then the set Knots ⁡ S {\displaystyle \operatorname {Knots} \mathbb {S} } of all knots of all strings in S {\displaystyle \mathbb {S} } forms

1665-455: A given topological field K {\displaystyle \mathbb {K} } is commonly denoted T V S K {\displaystyle \mathrm {TVS} _{\mathbb {K} }} or T V e c t K . {\displaystyle \mathrm {TVect} _{\mathbb {K} }.} The objects are the topological vector spaces over K {\displaystyle \mathbb {K} } and

1776-987: A linear extension of f : S → Y {\displaystyle f:S\to Y} exists then the linear extension F : span ⁡ S → Y {\displaystyle F:\operatorname {span} S\to Y} is unique and F ( c 1 s 1 + ⋯ c n s n ) = c 1 f ( s 1 ) + ⋯ + c n f ( s n ) {\displaystyle F\left(c_{1}s_{1}+\cdots c_{n}s_{n}\right)=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right)} holds for all n , c 1 , … , c n , {\displaystyle n,c_{1},\ldots ,c_{n},} and s 1 , … , s n {\displaystyle s_{1},\ldots ,s_{n}} as above. If S {\displaystyle S}

1887-988: A linear map F : span ⁡ S → Y {\displaystyle F:\operatorname {span} S\to Y} if and only if whenever n > 0 {\displaystyle n>0} is an integer, c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} are scalars, and s 1 , … , s n ∈ S {\displaystyle s_{1},\ldots ,s_{n}\in S} are vectors such that 0 = c 1 s 1 + ⋯ + c n s n , {\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},} then necessarily 0 = c 1 f ( s 1 ) + ⋯ + c n f ( s n ) . {\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).} If

1998-684: A linear map is one which preserves linear combinations . Denoting the zero elements of the vector spaces V {\displaystyle V} and W {\displaystyle W} by 0 V {\textstyle \mathbf {0} _{V}} and 0 W {\textstyle \mathbf {0} _{W}} respectively, it follows that f ( 0 V ) = 0 W . {\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.} Let c = 0 {\displaystyle c=0} and v ∈ V {\textstyle \mathbf {v} \in V} in

2109-521: A linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a conformal linear transformation . The composition of linear maps is linear: if f : V → W {\displaystyle f:V\to W} and g : W → Z {\textstyle g:W\to Z} are linear, then so is their composition g ∘ f : V → Z {\textstyle g\circ f:V\to Z} . It follows from this that

2220-917: A linear map on span ⁡ { ( 1 , 0 ) , ( 0 , 1 ) } = R 2 . {\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.} The unique linear extension F : R 2 → R {\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} } is the map that sends ( x , y ) = x ( 1 , 0 ) + y ( 0 , 1 ) ∈ R 2 {\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}} to F ( x , y ) = x ( − 1 ) + y ( 2 ) = − x + 2 y . {\displaystyle F(x,y)=x(-1)+y(2)=-x+2y.} Every (scalar-valued) linear functional f {\displaystyle f} defined on

2331-492: A lower dimension ); for example, it maps a plane through the origin in V {\displaystyle V} to either a plane through the origin in W {\displaystyle W} , a line through the origin in W {\displaystyle W} , or just the origin in W {\displaystyle W} . Linear maps can often be represented as matrices , and simple examples include rotation and reflection linear transformations . In

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2442-481: A natural topological structure : the norm induces a metric and the metric induces a topology. This is a topological vector space because : Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces. There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions ,

2553-540: A neighborhood basis consisting of closed convex balanced neighborhoods of the origin. Bounded subsets A subset E {\displaystyle E} of a topological vector space X {\displaystyle X} is bounded if for every neighborhood V {\displaystyle V} of the origin there exists t {\displaystyle t} such that E ⊆ t V {\displaystyle E\subseteq tV} . The definition of boundedness can be weakened

2664-407: A one-dimensional vector space over itself is called a linear functional . These statements generalize to any left-module R M {\textstyle {}_{R}M} over a ring R {\displaystyle R} without modification, and to any right-module upon reversing of the scalar multiplication. Often, a linear map is constructed by defining it on a subset of

2775-414: A regional Silesia commercial DVB-T free-to-air television station TVS (Russia) , a defunct Russian television channel TV Syd , a Danish government-owned radio and television public broadcasting company TVS China, also known as Southern Television Guangdong , a regional television network TVS (Venezuela) , Venezuelan regional television channel based in the city of Maracay TVS, former name of

2886-655: A smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory , the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem . No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on

2997-540: A string beginning with U 1 = U . {\displaystyle U_{1}=U.} This is called the natural string of U {\displaystyle U} Moreover, if a vector space X {\displaystyle X} has countable dimension then every string contains an absolutely convex string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of

3108-527: A topological vector space X {\displaystyle X} (that is probably not Hausdorff), form the quotient space X / M {\displaystyle X/M} where M {\displaystyle M} is the closure of { 0 } . {\displaystyle \{0\}.} X / M {\displaystyle X/M} is then a Hausdorff topological vector space that can be studied instead of X . {\displaystyle X.} One of

3219-403: A topological vector space. Given a subspace M ⊆ X , {\displaystyle M\subseteq X,} the quotient space X / M {\displaystyle X/M} with the usual quotient topology is a Hausdorff topological vector space if and only if M {\displaystyle M} is closed. This permits the following construction: given

3330-392: A topology to form a vector topology. Since every vector topology is translation invariant (which means that for all x 0 ∈ X , {\displaystyle x_{0}\in X,} the map X → X {\displaystyle X\to X} defined by x ↦ x 0 + x {\displaystyle x\mapsto x_{0}+x}

3441-536: A type of krytron Trinity Valley School , a private school in Fort Worth, Texas The Virgin Suicides , a 1993 novel Tornado vortex signature T. V. Sankaranarayanan , Indian singer Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title TVS . If an internal link led you here, you may wish to change the link to point directly to

TVS - Misplaced Pages Continue

3552-500: A vector ( a , b ), the value of a is the obstruction to there being a solution. An example illustrating the infinite-dimensional case is afforded by the map f : R → R , { a n } ↦ { b n } {\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}} with b 1 = 0 and b n + 1 = a n for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of

3663-514: A vector space X {\displaystyle X} and if s {\displaystyle s} is a scalar, then by definition: If S {\displaystyle \mathbb {S} } is a collection sequences of subsets of X , {\displaystyle X,} then S {\displaystyle \mathbb {S} } is said to be directed ( downwards ) under inclusion or simply directed downward if S {\displaystyle \mathbb {S} }

3774-424: A vector space and then extending by linearity to the linear span of the domain. Suppose X {\displaystyle X} and Y {\displaystyle Y} are vector spaces and f : S → Y {\displaystyle f:S\to Y} is a function defined on some subset S ⊆ X . {\displaystyle S\subseteq X.} Then

3885-2153: A vector space such that 0 ∈ U i {\displaystyle 0\in U_{i}} and U i + 1 + U i + 1 ⊆ U i {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}} for all i ≥ 0. {\displaystyle i\geq 0.} For all u ∈ U 0 , {\displaystyle u\in U_{0},} let S ( u ) := { n ∙ = ( n 1 , … , n k )   :   k ≥ 1 , n i ≥ 0  for all  i ,  and  u ∈ U n 1 + ⋯ + U n k } . {\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.} Define f : X → [ 0 , 1 ] {\displaystyle f:X\to [0,1]} by f ( x ) = 1 {\displaystyle f(x)=1} if x ∉ U 0 {\displaystyle x\not \in U_{0}} and otherwise let f ( x ) := inf { 2 − n 1 + ⋯ 2 − n k   :   n ∙ = ( n 1 , … , n k ) ∈ S ( x ) } . {\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.} Then f {\displaystyle f}

3996-399: Is not a filter base then it will form a neighborhood sub basis at 0 {\displaystyle 0} (rather than a neighborhood basis) for a vector topology on X . {\displaystyle X.} In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at

4107-411: Is isomorphic to the general linear group GL ⁡ ( n , K ) {\textstyle \operatorname {GL} (n,K)} of all n × n {\textstyle n\times n} invertible matrices with entries in K {\textstyle K} . If f : V → W {\textstyle f:V\to W} is linear, we define

4218-612: Is a function space , which is a common convention in functional analysis . Sometimes the term linear function has the same meaning as linear map , while in analysis it does not. A linear map from V {\displaystyle V} to W {\displaystyle W} always maps the origin of V {\displaystyle V} to the origin of W {\displaystyle W} . Moreover, it maps linear subspaces in V {\displaystyle V} onto linear subspaces in W {\displaystyle W} (possibly of

4329-522: Is a group (as all vector spaces are), τ {\displaystyle \tau } is a topology on X , {\displaystyle X,} and X × X {\displaystyle X\times X} is endowed with the product topology , then the addition map X × X → X {\displaystyle X\times X\to X} (defined by ( x , y ) ↦ x + y {\displaystyle (x,y)\mapsto x+y} )

4440-561: Is a homeomorphism ), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin. Theorem   (Neighborhood filter of the origin)  —  Suppose that X {\displaystyle X} is a real or complex vector space. If B {\displaystyle {\mathcal {B}}} is a non-empty additive collection of balanced and absorbing subsets of X {\displaystyle X} then B {\displaystyle {\mathcal {B}}}

4551-419: Is a neighborhood (resp. open neighborhood, closed neighborhood) of x {\displaystyle x} in X {\displaystyle X} if and only if the same is true of S {\displaystyle S} at the origin. A subset E {\displaystyle E} of a vector space X {\displaystyle X} is said to be Every neighborhood of

TVS - Misplaced Pages Continue

4662-422: Is a neighborhood base at 0 {\displaystyle 0} for a vector topology on X . {\displaystyle X.} That is, the assumptions are that B {\displaystyle {\mathcal {B}}} is a filter base that satisfies the following conditions: If B {\displaystyle {\mathcal {B}}} satisfies the above two conditions but

4773-456: Is a prefilter with respect to the containment ⊆ {\displaystyle \,\subseteq \,} defined above). Notation : Let Knots ⁡ S := ⋃ U ∙ ∈ S Knots ⁡ U ∙ {\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }} be

4884-698: Is a subspace of V {\textstyle V} and im ⁡ ( f ) {\textstyle \operatorname {im} (f)} is a subspace of W {\textstyle W} . The following dimension formula is known as the rank–nullity theorem : dim ⁡ ( ker ⁡ ( f ) ) + dim ⁡ ( im ⁡ ( f ) ) = dim ⁡ ( V ) . {\displaystyle \dim(\ker(f))+\dim(\operatorname {im} (f))=\dim(V).} The number dim ⁡ ( im ⁡ ( f ) ) {\textstyle \dim(\operatorname {im} (f))}

4995-880: Is a surjective TVS embedding Many properties of TVSs that are studied, such as local convexity , metrizability , completeness , and normability , are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection N {\displaystyle {\mathcal {N}}} of subsets of a vector space is called additive if for every N ∈ N , {\displaystyle N\in {\mathcal {N}},} there exists some U ∈ N {\displaystyle U\in {\mathcal {N}}} such that U + U ⊆ N . {\displaystyle U+U\subseteq N.} Characterization of continuity of addition at 0 {\displaystyle 0}  —  If ( X , + ) {\displaystyle (X,+)}

5106-595: Is a vector space over a topological field K {\displaystyle \mathbb {K} } (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition ⋅ + ⋅ : X × X → X {\displaystyle \cdot \,+\,\cdot \;:X\times X\to X} and scalar multiplication ⋅ : K × X → X {\displaystyle \cdot :\mathbb {K} \times X\to X} are continuous functions (where

5217-937: Is a homeomorphism. Using s = − 1 {\displaystyle s=-1} produces the negation map X → X {\displaystyle X\to X} defined by x ↦ − x , {\displaystyle x\mapsto -x,} which is consequently a linear homeomorphism and thus a TVS-isomorphism. If x ∈ X {\displaystyle x\in X} and any subset S ⊆ X , {\displaystyle S\subseteq X,} then cl X ⁡ ( x + S ) = x + cl X ⁡ S {\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S} and moreover, if 0 ∈ S {\displaystyle 0\in S} then x + S {\displaystyle x+S}

5328-510: Is a linear map, f ( v ) = f ( c 1 v 1 + ⋯ + c n v n ) = c 1 f ( v 1 ) + ⋯ + c n f ( v n ) , {\displaystyle f(\mathbf {v} )=f(c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n})=c_{1}f(\mathbf {v} _{1})+\cdots +c_{n}f\left(\mathbf {v} _{n}\right),} which implies that

5439-404: Is a linear map. In particular, if f {\displaystyle f} has a linear extension to span ⁡ S , {\displaystyle \operatorname {span} S,} then it has a linear extension to all of X . {\displaystyle X.} The map f : S → Y {\displaystyle f:S\to Y} can be extended to

5550-406: Is a topological vector space and if all U i {\displaystyle U_{i}} are neighborhoods of the origin then f {\displaystyle f} is continuous, where if in addition X {\displaystyle X} is Hausdorff and U ∙ {\displaystyle U_{\bullet }} forms a basis of balanced neighborhoods of

5661-465: Is a topological vector space then there exists a set S {\displaystyle \mathbb {S} } of neighborhood strings in X {\displaystyle X} that is directed downward and such that the set of all knots of all strings in S {\displaystyle \mathbb {S} } is a neighborhood basis at the origin for ( X , τ ) . {\displaystyle (X,\tau ).} Such

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5772-448: Is a vector ( a 1 j ⋮ a m j ) {\displaystyle {\begin{pmatrix}a_{1j}\\\vdots \\a_{mj}\end{pmatrix}}} corresponding to f ( v j ) {\displaystyle f(\mathbf {v} _{j})} as defined above. To define it more clearly, for some column j {\displaystyle j} that corresponds to

5883-392: Is also called the rank of f {\textstyle f} and written as rank ⁡ ( f ) {\textstyle \operatorname {rank} (f)} , or sometimes, ρ ( f ) {\textstyle \rho (f)} ; the number dim ⁡ ( ker ⁡ ( f ) ) {\textstyle \dim(\ker(f))}

5994-454: Is also linear. Thus the set L ( V , W ) {\textstyle {\mathcal {L}}(V,W)} of linear maps from V {\textstyle V} to W {\textstyle W} itself forms a vector space over K {\textstyle K} , sometimes denoted Hom ⁡ ( V , W ) {\textstyle \operatorname {Hom} (V,W)} . Furthermore, in

6105-406: Is an endomorphism of V {\textstyle V} ; the set of all such endomorphisms End ⁡ ( V ) {\textstyle \operatorname {End} (V)} together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K {\textstyle K} (and in particular

6216-560: Is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication. By the associativity of the addition operation denoted as +, for any vectors u 1 , … , u n ∈ V {\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V} and scalars c 1 , … , c n ∈ K , {\textstyle c_{1},\ldots ,c_{n}\in K,}

6327-510: Is called the beginning of U ∙ . {\displaystyle U_{\bullet }.} The sequence U ∙ {\displaystyle U_{\bullet }} is/is a: If U {\displaystyle U} is an absorbing disk in a vector space X {\displaystyle X} then the sequence defined by U i := 2 1 − i U {\displaystyle U_{i}:=2^{1-i}U} forms

6438-438: Is called the nullity of f {\textstyle f} and written as null ⁡ ( f ) {\textstyle \operatorname {null} (f)} or ν ( f ) {\textstyle \nu (f)} . If V {\textstyle V} and W {\textstyle W} are finite-dimensional, bases have been chosen and f {\textstyle f}

6549-410: Is continuous at the origin of X × X {\displaystyle X\times X} if and only if the set of neighborhoods of the origin in ( X , τ ) {\displaystyle (X,\tau )} is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood." All of the above conditions are consequently a necessity for

6660-414: Is defined as coker ⁡ ( f ) := W / f ( V ) = W / im ⁡ ( f ) . {\displaystyle \operatorname {coker} (f):=W/f(V)=W/\operatorname {im} (f).} This is the dual notion to the kernel: just as the kernel is a sub space of the domain, the co-kernel is a quotient space of the target. Formally, one has

6771-422: Is linear and α {\textstyle \alpha } is an element of the ground field K {\textstyle K} , then the map α f {\textstyle \alpha f} , defined by ( α f ) ( x ) = α ( f ( x ) ) {\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))} ,

SECTION 60

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6882-521: Is linearly independent then every function f : S → Y {\displaystyle f:S\to Y} into any vector space has a linear extension to a (linear) map span ⁡ S → Y {\displaystyle \;\operatorname {span} S\to Y} (the converse is also true). For example, if X = R 2 {\displaystyle X=\mathbb {R} ^{2}} and Y = R {\displaystyle Y=\mathbb {R} } then

6993-404: Is locally convex, the boundedness can be characterized by seminorms : the subset E {\displaystyle E} is bounded if and only if every continuous seminorm p {\displaystyle p} is bounded on E . {\displaystyle E.} Linear map In mathematics , and more specifically in linear algebra , a linear map (also called

7104-703: Is not empty and for all U ∙ , V ∙ ∈ S , {\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,} there exists some W ∙ ∈ S {\displaystyle W_{\bullet }\in \mathbb {S} } such that W ∙ ⊆ U ∙ {\displaystyle W_{\bullet }\subseteq U_{\bullet }} and W ∙ ⊆ V ∙ {\displaystyle W_{\bullet }\subseteq V_{\bullet }} (said differently, if and only if S {\displaystyle \mathbb {S} }

7215-404: Is represented by the matrix A {\textstyle A} , then the rank and nullity of f {\textstyle f} are equal to the rank and nullity of the matrix A {\textstyle A} , respectively. A subtler invariant of a linear transformation f : V → W {\textstyle f:V\to W} is the co kernel , which

7326-451: Is said to be a linear map if for any two vectors u , v ∈ V {\textstyle \mathbf {u} ,\mathbf {v} \in V} and any scalar c ∈ K {\displaystyle c\in K} the following two conditions are satisfied: Thus, a linear map is said to be operation preserving . In other words, it does not matter whether the linear map

7437-1244: Is subadditive (meaning f ( x + y ) ≤ f ( x ) + f ( y ) {\displaystyle f(x+y)\leq f(x)+f(y)} for all x , y ∈ X {\displaystyle x,y\in X} ) and f = 0 {\displaystyle f=0} on ⋂ i ≥ 0 U i ; {\textstyle \bigcap _{i\geq 0}U_{i};} so in particular, f ( 0 ) = 0. {\displaystyle f(0)=0.} If all U i {\displaystyle U_{i}} are symmetric sets then f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} and if all U i {\displaystyle U_{i}} are balanced then f ( s x ) ≤ f ( x ) {\displaystyle f(sx)\leq f(x)} for all scalars s {\displaystyle s} such that | s | ≤ 1 {\displaystyle |s|\leq 1} and all x ∈ X . {\displaystyle x\in X.} If X {\displaystyle X}

7548-599: Is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1. For a linear operator with finite-dimensional kernel and co-kernel, one may define index as: ind ⁡ ( f ) := dim ⁡ ( ker ⁡ ( f ) ) − dim ⁡ ( coker ⁡ ( f ) ) , {\displaystyle \operatorname {ind} (f):=\dim(\ker(f))-\dim(\operatorname {coker} (f)),} namely

7659-595: Is the group of units in the ring End ⁡ ( V ) {\textstyle \operatorname {End} (V)} . If V {\textstyle V} has finite dimension n {\textstyle n} , then End ⁡ ( V ) {\textstyle \operatorname {End} (V)} is isomorphic to the associative algebra of all n × n {\textstyle n\times n} matrices with entries in K {\textstyle K} . The automorphism group of V {\textstyle V}

7770-463: Is the matrix of f {\displaystyle f} . In other words, every column j = 1 , … , n {\displaystyle j=1,\ldots ,n} has a corresponding vector f ( v j ) {\displaystyle f(\mathbf {v} _{j})} whose coordinates a 1 j , ⋯ , a m j {\displaystyle a_{1j},\cdots ,a_{mj}} are

7881-417: Is the set of all topological strings in a TVS ( X , τ ) {\displaystyle (X,\tau )} then τ S = τ . {\displaystyle \tau _{\mathbb {S} }=\tau .} A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string. A vector space is an abelian group with respect to

7992-474: Is their pointwise sum f 1 + f 2 {\displaystyle f_{1}+f_{2}} , which is defined by ( f 1 + f 2 ) ( x ) = f 1 ( x ) + f 2 ( x ) {\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )} . If f : V → W {\textstyle f:V\to W}

8103-532: The Schwartz spaces , and spaces of test functions and the spaces of distributions on them. These are all examples of Montel spaces . An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion . A topological field is a topological vector space over each of its subfields . A topological vector space ( TVS ) X {\displaystyle X}

8214-411: The class of all vector spaces over a given field K , together with K -linear maps as morphisms , forms a category . The inverse of a linear map, when defined, is again a linear map. If f 1 : V → W {\textstyle f_{1}:V\to W} and f 2 : V → W {\textstyle f_{2}:V\to W} are linear, then so

8325-410: The exact sequence 0 → ker ⁡ ( f ) → V → W → coker ⁡ ( f ) → 0. {\displaystyle 0\to \ker(f)\to V\to W\to \operatorname {coker} (f)\to 0.} These can be interpreted thus: given a linear equation f ( v ) = w to solve, The dimension of the co-kernel and the dimension of

8436-882: The kernel and the image or range of f {\textstyle f} by ker ⁡ ( f ) = { x ∈ V : f ( x ) = 0 } im ⁡ ( f ) = { w ∈ W : w = f ( x ) , x ∈ V } {\displaystyle {\begin{aligned}\ker(f)&=\{\,\mathbf {x} \in V:f(\mathbf {x} )=\mathbf {0} \,\}\\\operatorname {im} (f)&=\{\,\mathbf {w} \in W:\mathbf {w} =f(\mathbf {x} ),\mathbf {x} \in V\,\}\end{aligned}}} ker ⁡ ( f ) {\textstyle \ker(f)}

8547-442: The morphisms are the continuous K {\displaystyle \mathbb {K} } -linear maps from one object to another. A topological vector space homomorphism (abbreviated TVS homomorphism ), also called a topological homomorphism , is a continuous linear map u : X → Y {\displaystyle u:X\to Y} between topological vector spaces (TVSs) such that

8658-703: The "longer" method going clockwise from the same point such that [ v ] B ′ {\textstyle \left[\mathbf {v} \right]_{B'}} is left-multiplied with P − 1 A P {\textstyle P^{-1}AP} , or P − 1 A P [ v ] B ′ = [ T ( v ) ] B ′ {\textstyle P^{-1}AP\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}} . In two- dimensional space R linear maps are described by 2 × 2 matrices . These are some examples: If

8769-813: The Brazilian channel Sistema Brasileiro de Televisão Television Saitama , Japan TVS (São Tomé and Príncipe) , the public television broadcaster of São Tomé and Príncipe TVS (Malaysian TV channel) , a Malaysian television channel Other [ edit ] T. V. Sundram Iyengar , Indian industrialist TVS Group , an industrial conglomerate based in India TVS Electronics , computer peripherals manufacturing company TVS Motor , motor manufacturing company IATA code for Tangshan Sannühe Airport Transvaginal ultrasound Transient voltage suppressor , an electronic component used for surge protection Triggered vacuum switch ,

8880-425: The assignment ( 1 , 0 ) → − 1 {\displaystyle (1,0)\to -1} and ( 0 , 1 ) → 2 {\displaystyle (0,1)\to 2} can be linearly extended from the linearly independent set of vectors S := { ( 1 , 0 ) , ( 0 , 1 ) } {\displaystyle S:=\{(1,0),(0,1)\}} to

8991-414: The basic properties of topological vector spaces. Theorem   ( R {\displaystyle \mathbb {R} } -valued function induced by a string)  —  Let U ∙ = ( U i ) i = 0 ∞ {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }} be a collection of subsets of

9102-512: The bottom right corner [ T ( v ) ] B ′ {\textstyle \left[T\left(\mathbf {v} \right)\right]_{B'}} , one would left-multiply—that is, A ′ [ v ] B ′ = [ T ( v ) ] B ′ {\textstyle A'\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}} . The equivalent method would be

9213-424: The case that V = W {\textstyle V=W} , this vector space, denoted End ⁡ ( V ) {\textstyle \operatorname {End} (V)} , is an associative algebra under composition of maps , since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below. Given again

9324-497: The classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ℵ 0 + 0 = ℵ 0 + 1 {\textstyle \aleph _{0}+0=\aleph _{0}+1} ), but in

9435-498: The codomain of f . {\displaystyle f.} When the subset S {\displaystyle S} is a vector subspace of X {\displaystyle X} then a ( Y {\displaystyle Y} -valued) linear extension of f {\displaystyle f} to all of X {\displaystyle X} is guaranteed to exist if (and only if) f : S → Y {\displaystyle f:S\to Y}

9546-428: The degrees of freedom minus the number of constraints. For a transformation between finite-dimensional vector spaces, this is just the difference dim( V ) − dim( W ), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from

9657-556: The domain of f {\displaystyle f} ) then there exists a linear extension to X {\displaystyle X} that is also dominated by p . {\displaystyle p.} If V {\displaystyle V} and W {\displaystyle W} are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V {\displaystyle V} to W {\displaystyle W} can be represented by

9768-488: The domains of these functions are endowed with product topologies ). Such a topology is called a vector topology or a TVS topology on X . {\displaystyle X.} Every topological vector space is also a commutative topological group under addition. Hausdorff assumption Many authors (for example, Walter Rudin ), but not this page, require the topology on X {\displaystyle X} to be T 1 ; it then follows that

9879-462: The elements of column j {\displaystyle j} . A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen. The matrices of a linear transformation can be represented visually: Such that starting in the bottom left corner [ v ] B ′ {\textstyle \left[\mathbf {v} \right]_{B'}} and looking for

9990-433: The equation for homogeneity of degree 1: f ( 0 V ) = f ( 0 v ) = 0 f ( v ) = 0 W . {\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.} A linear map V → K {\displaystyle V\to K} with K {\displaystyle K} viewed as

10101-408: The finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication , the addition of linear maps corresponds to the matrix addition , and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars. A linear transformation f : V → V {\textstyle f:V\to V}

10212-450: The following equality holds: f ( c 1 u 1 + ⋯ + c n u n ) = c 1 f ( u 1 ) + ⋯ + c n f ( u n ) . {\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).} Thus

10323-655: The function f {\displaystyle f} is entirely determined by the values of a i j {\displaystyle a_{ij}} . If we put these values into an m × n {\displaystyle m\times n} matrix M {\displaystyle M} , then we can conveniently use it to compute the vector output of f {\displaystyle f} for any vector in V {\displaystyle V} . To get M {\displaystyle M} , every column j {\displaystyle j} of M {\displaystyle M}

10434-412: The function f is entirely determined by the vectors f ( v 1 ) , … , f ( v n ) {\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})} . Now let { w 1 , … , w m } {\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}} be

10545-398: The image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W / f ( V ) is the dimension of the target space minus the dimension of the image. As a simple example, consider the map f : R → R , given by f ( x , y ) = (0, y ). Then for an equation f ( x , y ) = ( a , b ) to have a solution, we must have

10656-557: The induced map u : X → Im ⁡ u {\displaystyle u:X\to \operatorname {Im} u} is an open mapping when Im ⁡ u := u ( X ) , {\displaystyle \operatorname {Im} u:=u(X),} which is the range or image of u , {\displaystyle u,} is given the subspace topology induced by Y . {\displaystyle Y.} A topological vector space embedding (abbreviated TVS embedding ), also called

10767-403: The infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h : R → R , { a n } ↦ { c n } {\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}} with c n = a n + 1 . Its image

10878-460: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=TVS&oldid=1183610651 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Topological vector space Many topological vector spaces are spaces of functions , or linear operators acting on topological vector spaces, and

10989-421: The language of category theory , linear maps are the morphisms of vector spaces, and they form a category equivalent to the one of matrices . Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the same field K {\displaystyle K} . A function f : V → W {\displaystyle f:V\to W}

11100-486: The mapping f ( v j ) {\displaystyle f(\mathbf {v} _{j})} , M = (   ⋯ a 1 j ⋯   ⋮ a m j ) {\displaystyle \mathbf {M} ={\begin{pmatrix}\ \cdots &a_{1j}&\cdots \ \\&\vdots &\\&a_{mj}&\end{pmatrix}}} where M {\displaystyle M}

11211-436: The most used properties of vector topologies is that every vector topology is translation invariant : Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if s ≠ 0 {\displaystyle s\neq 0} then the linear map X → X {\displaystyle X\to X} defined by x ↦ s x {\displaystyle x\mapsto sx}

11322-403: The operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by − 1 {\displaystyle -1} ). Hence, every topological vector space is an abelian topological group . Every TVS is completely regular but a TVS need not be normal . Let X {\displaystyle X} be

11433-399: The origin for any vector topology. Let X {\displaystyle X} be a vector space and let U ∙ = ( U i ) i = 1 ∞ {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }} be a sequence of subsets of X . {\displaystyle X.} Each set in

11544-807: The origin in X {\displaystyle X} then d ( x , y ) := f ( x − y ) {\displaystyle d(x,y):=f(x-y)} is a metric defining the vector topology on X . {\displaystyle X.} A proof of the above theorem is given in the article on metrizable topological vector spaces . If U ∙ = ( U i ) i ∈ N {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }} and V ∙ = ( V i ) i ∈ N {\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }} are two collections of subsets of

11655-402: The origin is an absorbing set and contains an open balanced neighborhood of 0 {\displaystyle 0} so every topological vector space has a local base of absorbing and balanced sets . The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0 ; {\displaystyle 0;} if the space is locally convex then it also has

11766-550: The sequence U ∙ {\displaystyle U_{\bullet }} is called a knot of U ∙ {\displaystyle U_{\bullet }} and for every index i , {\displaystyle i,} U i {\displaystyle U_{i}} is called the i {\displaystyle i} -th knot of U ∙ . {\displaystyle U_{\bullet }.} The set U 1 {\displaystyle U_{1}}

11877-563: The set of all automorphisms of V {\textstyle V} forms a group , the automorphism group of V {\textstyle V} which is denoted by Aut ⁡ ( V ) {\textstyle \operatorname {Aut} (V)} or GL ⁡ ( V ) {\textstyle \operatorname {GL} (V)} . Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut ⁡ ( V ) {\textstyle \operatorname {Aut} (V)}

11988-405: The set of all knots of all strings in S . {\displaystyle \mathbb {S} .} Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. Theorem   (Topology induced by strings)  —  If ( X , τ ) {\displaystyle (X,\tau )}

12099-417: The space is Hausdorff , and even Tychonoff . A topological vector space is said to be separated if it is Hausdorff; importantly, "separated" does not mean separable . The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below . Category and morphisms The category of topological vector spaces over

12210-447: The topology is often defined so as to capture a particular notion of convergence of sequences of functions. In this article, the scalar field of a topological vector space will be assumed to be either the complex numbers C {\displaystyle \mathbb {C} } or the real numbers R , {\displaystyle \mathbb {R} ,} unless clearly stated otherwise. Every normed vector space has

12321-426: The vector space. Let V and W denote vector spaces over a field F and let T : V → W be a linear map. T is said to be injective or a monomorphism if any of the following equivalent conditions are true: T is said to be surjective or an epimorphism if any of the following equivalent conditions are true: T is said to be an isomorphism if it is both left- and right-invertible. This

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