The smallest such is called the operator norm of and denoted by A bounded operator between normed spaces is continuous and vice versa.
The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
Outside of functional analysis, when a function is called "bounded" then this usually means that its image is a bounded subset of its codomain. A linear map has this property if and only if it is identically Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).
A linear operator between normed spaces is bounded if and only if it is continuous.
Proof
Suppose that is bounded. Then, for all vectors with nonzero we have
Letting go to zero shows that is continuous at Moreover, since the constant does not depend on this shows that in fact is uniformly continuous, and even Lipschitz continuous.
Conversely, it follows from the continuity at the zero vector that there exists a such that for all vectors with Thus, for all non-zero one has
A linear operator between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then is bounded in A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.
Continuity and boundedness
Every sequentially continuous linear operator between TVS is a bounded operator.[1] This implies that every continuous linear operator between metrizable TVS is bounded. However, in general, a bounded linear operator between two TVSs need not be continuous.
This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.
If is a linear operator between two topological vector spaces and if there exists a neighborhood of the origin in such that is a bounded subset of then is continuous.[2] This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous. In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a normed space).
Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous. That is, a locally convex TVS is a bornological space if and only if for every locally convex TVS a linear operator is continuous if and only if it is bounded.[3]
Every normed space is bornological.
Characterizations of bounded linear operators
Let be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent:
(Definition): maps bounded subsets of its domain to bounded subsets of its codomain;[3]
maps bounded subsets of its domain to bounded subsets of its image;[3]
maps every null sequence to a bounded sequence;[3]
A null sequence is by definition a sequence that converges to the origin.
Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map.
maps every Mackey convergent null sequence to a bounded subset of [note 1]
A sequence is said to be Mackey convergent to the origin in if there exists a divergent sequence of positive real number such that is a bounded subset of
if and are locally convex then the following may be add to this list:
A sequentially continuous linear map between two TVSs is always bounded,[1] but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex).
Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
Any linear operator defined on a finite-dimensional normed space is bounded.
On the sequence space of eventually zero sequences of real numbers, considered with the norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the norm, the same operator is not bounded.
is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula
is bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators.
Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback
^Proof: Assume for the sake of contradiction that converges to but is not bounded in Pick an open balanced neighborhood of the origin in such that does not absorb the sequence Replacing with a subsequence if necessary, it may be assumed without loss of generality that for every positive integer The sequence is Mackey convergent to the origin (since is bounded in ) so by assumption, is bounded in So pick a real such that for every integer If is an integer then since is balanced, which is a contradiction. Q.E.D. This proof readily generalizes to give even stronger characterizations of " is bounded." For example, the word "such that is a bounded subset of " in the definition of "Mackey convergent to the origin" can be replaced with "such that in "
bounded, operator, confused, with, bounded, function, theory, functional, analysis, operator, theory, bounded, linear, operator, linear, transformation, displaystyle, between, topological, vector, spaces, tvss, displaystyle, displaystyle, that, maps, bounded, . Not to be confused with bounded function set theory In functional analysis and operator theory a bounded linear operator is a linear transformation L X Y displaystyle L X to Y between topological vector spaces TVSs X displaystyle X and Y displaystyle Y that maps bounded subsets of X displaystyle X to bounded subsets of Y displaystyle Y If X displaystyle X and Y displaystyle Y are normed vector spaces a special type of TVS then L displaystyle L is bounded if and only if there exists some M gt 0 displaystyle M gt 0 such that for all x X displaystyle x in X L x Y M x X displaystyle Lx Y leq M x X The smallest such M displaystyle M is called the operator norm of L displaystyle L and denoted by L displaystyle L A bounded operator between normed spaces is continuous and vice versa The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces Outside of functional analysis when a function f X Y displaystyle f X to Y is called bounded then this usually means that its image f X displaystyle f X is a bounded subset of its codomain A linear map has this property if and only if it is identically 0 displaystyle 0 Consequently in functional analysis when a linear operator is called bounded then it is never meant in this abstract sense of having a bounded image Contents 1 In normed vector spaces 1 1 Equivalence of boundedness and continuity 2 In topological vector spaces 2 1 Continuity and boundedness 2 1 1 Bornological spaces 2 2 Characterizations of bounded linear operators 3 Examples 3 1 Unbounded linear operators 3 2 Properties of the space of bounded linear operators 4 See also 5 References 6 BibliographyIn normed vector spaces EditEvery bounded operator is Lipschitz continuous at 0 displaystyle 0 Equivalence of boundedness and continuity Edit A linear operator between normed spaces is bounded if and only if it is continuous Proof Suppose that L displaystyle L is bounded Then for all vectors x h X displaystyle x h in X with h displaystyle h nonzero we have L x h L x L h M h displaystyle L x h L x L h leq M h Letting h displaystyle h go to zero shows that L displaystyle L is continuous at x displaystyle x Moreover since the constant M displaystyle M does not depend on x displaystyle x this shows that in fact L displaystyle L is uniformly continuous and even Lipschitz continuous Conversely it follows from the continuity at the zero vector that there exists a d gt 0 displaystyle delta gt 0 such that L h L h L 0 1 displaystyle L h L h L 0 leq 1 for all vectors h X displaystyle h in X with h d displaystyle h leq delta Thus for all non zero x X displaystyle x in X one has L x x d L d x x x d L d x x x d 1 1 d x displaystyle Lx left Vert x over delta L left delta x over x right right Vert x over delta left Vert L left delta x over x right right Vert leq x over delta cdot 1 1 over delta x This proves that L displaystyle L is bounded Q E D In topological vector spaces EditA linear operator F X Y displaystyle F X to Y between two topological vector spaces TVSs is called a bounded linear operator or just bounded if whenever B X displaystyle B subseteq X is bounded in X displaystyle X then F B displaystyle F B is bounded in Y displaystyle Y A subset of a TVS is called bounded or more precisely von Neumann bounded if every neighborhood of the origin absorbs it In a normed space and even in a seminormed space a subset is von Neumann bounded if and only if it is norm bounded Hence for normed spaces the notion of a von Neumann bounded set is identical to the usual notion of a norm bounded subset Continuity and boundedness Edit Every sequentially continuous linear operator between TVS is a bounded operator 1 This implies that every continuous linear operator between metrizable TVS is bounded However in general a bounded linear operator between two TVSs need not be continuous This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets In this context it is still true that every continuous map is bounded however the converse fails a bounded operator need not be continuous This also means that boundedness is no longer equivalent to Lipschitz continuity in this context If the domain is a bornological space for example a pseudometrizable TVS a Frechet space a normed space then a linear operators into any other locally convex spaces is bounded if and only if it is continuous For LF spaces a weaker converse holds any bounded linear map from an LF space is sequentially continuous If F X Y displaystyle F X to Y is a linear operator between two topological vector spaces and if there exists a neighborhood U displaystyle U of the origin in X displaystyle X such that F U displaystyle F U is a bounded subset of Y displaystyle Y then F displaystyle F is continuous 2 This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous In particular any linear functional that is bounded on some neighborhood of the origin is continuous even if its domain is not a normed space Bornological spaces Edit Main article Bornological space Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous That is a locally convex TVS X displaystyle X is a bornological space if and only if for every locally convex TVS Y displaystyle Y a linear operator F X Y displaystyle F X to Y is continuous if and only if it is bounded 3 Every normed space is bornological Characterizations of bounded linear operators Edit Let F X Y displaystyle F X to Y be a linear operator between topological vector spaces not necessarily Hausdorff The following are equivalent F displaystyle F is locally bounded 3 Definition F displaystyle F maps bounded subsets of its domain to bounded subsets of its codomain 3 F displaystyle F maps bounded subsets of its domain to bounded subsets of its image Im F F X displaystyle operatorname Im F F X 3 F displaystyle F maps every null sequence to a bounded sequence 3 A null sequence is by definition a sequence that converges to the origin Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map F displaystyle F maps every Mackey convergent null sequence to a bounded subset of Y displaystyle Y note 1 A sequence x x i i 1 displaystyle x bullet left x i right i 1 infty is said to be Mackey convergent to the origin in X displaystyle X if there exists a divergent sequence r r i i 1 displaystyle r bullet left r i right i 1 infty to infty of positive real number such that r r i x i i 1 displaystyle r bullet left r i x i right i 1 infty is a bounded subset of X displaystyle X if X displaystyle X and Y displaystyle Y are locally convex then the following may be add to this list F displaystyle F maps bounded disks into bounded disks 4 F 1 displaystyle F 1 maps bornivorous disks in Y displaystyle Y into bornivorous disks in X displaystyle X 4 if X displaystyle X is a bornological space and Y displaystyle Y is locally convex then the following may be added to this list F displaystyle F is sequentially continuous at some or equivalently at every point of its domain 5 A sequentially continuous linear map between two TVSs is always bounded 1 but the converse requires additional assumptions to hold such as the domain being bornological and the codomain being locally convex If the domain X displaystyle X is also a sequential space then F displaystyle F is sequentially continuous if and only if it is continuous F displaystyle F is sequentially continuous at the origin Examples EditAny linear operator between two finite dimensional normed spaces is bounded and such an operator may be viewed as multiplication by some fixed matrix Any linear operator defined on a finite dimensional normed space is bounded On the sequence space c 00 displaystyle c 00 of eventually zero sequences of real numbers considered with the ℓ 1 displaystyle ell 1 norm the linear operator to the real numbers which returns the sum of a sequence is bounded with operator norm 1 If the same space is considered with the ℓ displaystyle ell infty norm the same operator is not bounded Many integral transforms are bounded linear operators For instance if K a b c d R displaystyle K a b times c d to mathbb R is a continuous function then the operator L displaystyle L defined on the space C a b displaystyle C a b of continuous functions on a b displaystyle a b endowed with the uniform norm and with values in the space C c d displaystyle C c d with L displaystyle L given by the formula L f y a b K x y f x d x displaystyle Lf y int a b K x y f x dx is bounded This operator is in fact a compact operator The compact operators form an important class of bounded operators The Laplace operator D H 2 R n L 2 R n displaystyle Delta H 2 mathbb R n to L 2 mathbb R n its domain is a Sobolev space and it takes values in a space of square integrable functions is bounded The shift operator on the Lp space ℓ 2 displaystyle ell 2 of all sequences x 0 x 2 x 2 displaystyle left x 0 x 2 x 2 ldots right of real numbers with x 0 2 x 1 2 x 2 2 lt displaystyle x 0 2 x 1 2 x 2 2 cdots lt infty L x 0 x 1 x 2 0 x 0 x 1 x 2 displaystyle L x 0 x 1 x 2 dots left 0 x 0 x 1 x 2 ldots right is bounded Its operator norm is easily seen to be 1 displaystyle 1 Unbounded linear operators Edit Let X displaystyle X be the space of all trigonometric polynomials on p p displaystyle pi pi with the norm P p p P x d x displaystyle P int pi pi P x dx The operator L X X displaystyle L X to X that maps a polynomial to its derivative is not bounded Indeed for v n e i n x displaystyle v n e inx with n 1 2 displaystyle n 1 2 ldots we have v n 2 p displaystyle v n 2 pi while L v n 2 p n displaystyle L v n 2 pi n to infty as n displaystyle n to infty so L displaystyle L is not bounded Properties of the space of bounded linear operators Edit The space of all bounded linear operators from X displaystyle X to Y displaystyle Y is denoted by B X Y displaystyle B X Y and is a normed vector space If Y displaystyle Y is Banach then so is B X Y displaystyle B X Y from which it follows that dual spaces are Banach For any A B X Y displaystyle A in B X Y the kernel of A displaystyle A is a closed linear subspace of X displaystyle X If B X Y displaystyle B X Y is Banach and X displaystyle X is nontrivial then Y displaystyle Y is Banach See also EditBounded set topological vector space Generalization of boundedness Contraction operator theory Bounded operators with sub unit norm Discontinuous linear map Continuous linear operator Local boundedness Norm mathematics Length in a vector space Operator algebra Branch of functional analysis Operator norm Measure of the size of linear operators Operator theory Mathematical field of study Seminorm nonnegative real valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback Unbounded operator Linear operator defined on a dense linear subspaceReferences Edit Proof Assume for the sake of contradiction that x x i i 1 displaystyle x bullet left x i right i 1 infty converges to 0 displaystyle 0 but F x F x i i 1 displaystyle F left x bullet right left F left x i right right i 1 infty is not bounded in Y displaystyle Y Pick an open balanced neighborhood V displaystyle V of the origin in Y displaystyle Y such that V displaystyle V does not absorb the sequence F x displaystyle F left x bullet right Replacing x displaystyle x bullet with a subsequence if necessary it may be assumed without loss of generality that F x i i 2 V displaystyle F left x i right not in i 2 V for every positive integer i displaystyle i The sequence z x i i i 1 displaystyle z bullet left x i i right i 1 infty is Mackey convergent to the origin since i z i i 1 x i i 1 0 displaystyle left iz i right i 1 infty left x i right i 1 infty to 0 is bounded in X displaystyle X so by assumption F z F z i i 1 displaystyle F left z bullet right left F left z i right right i 1 infty is bounded in Y displaystyle Y So pick a real r gt 1 displaystyle r gt 1 such that F z i r V displaystyle F left z i right in rV for every integer i displaystyle i If i gt r displaystyle i gt r is an integer then since V displaystyle V is balanced F x i r i V i 2 V displaystyle F left x i right in riV subseteq i 2 V which is a contradiction Q E D This proof readily generalizes to give even stronger characterizations of F displaystyle F is bounded For example the word such that r i x i i 1 displaystyle left r i x i right i 1 infty is a bounded subset of X displaystyle X in the definition of Mackey convergent to the origin can be replaced with such that r i x i i 1 0 displaystyle left r i x i right i 1 infty to 0 in X displaystyle X a b Wilansky 2013 pp 47 50 Narici amp Beckenstein 2011 pp 156 175 a b c d e Narici amp Beckenstein 2011 pp 441 457 a b Narici amp Beckenstein 2011 p 444 Narici amp Beckenstein 2011 pp 451 457 Bibliography Edit Bounded operator Encyclopedia of Mathematics EMS Press 2001 1994 Kreyszig Erwin Introductory Functional Analysis with Applications Wiley 1989 Narici Lawrence Beckenstein Edward 2011 Topological Vector Spaces Pure and applied mathematics Second ed Boca Raton FL CRC Press ISBN 978 1584888666 OCLC 144216834 Wilansky Albert 2013 Modern Methods in Topological Vector Spaces Mineola New York Dover Publications Inc ISBN 978 0 486 49353 4 OCLC 849801114 Retrieved from https en wikipedia org w index php title Bounded operator amp oldid 1139180246, wikipedia, wiki, book, books, library,