{"history":[],"article":{"id":"da71556b-1c8b-4465-b71b-9ec6c6c0cde0","title":"Bounded operator","slug":"bounded-operator","content":":::summary\n:::\n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory), a **[bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) linear operator** is a [linear transformation](https://bliptext.com/articles/linear-transformation) L : X \\to Y between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) X and Y that maps [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subsets of X to [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subsets of Y. If X and Y are [normed vector spaces](https://bliptext.com/articles/normed-vector-space) (a special type of TVS), then L is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) if and only if there exists some M > 0 such that for all x \\in X,\nThe smallest such M is called the [operator norm](https://bliptext.com/articles/operator-norm) of L and denoted by \\|L\\|.\nA bounded operator between normed spaces is [continuous](https://bliptext.com/articles/continuous-linear-operator) and vice versa.\nThe concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.\nOutside of functional analysis, when a function f : X \\to Y is called \"[bounded](https://bliptext.com/articles/bounded-function)\" then this usually means that its [image](https://bliptext.com/articles/image-of-a-function) f(X) is a [bounded](https://bliptext.com/articles/bounded-function) subset of its codomain.\nA linear map has this property if and only if it is identically 0.\nConsequently, in functional analysis, when a linear operator is called \"bounded\" then it is never meant in this abstract sense (of having a bounded image).\n\n# In normed vector spaces\nEvery bounded operator is [Lipschitz continuous](https://bliptext.com/articles/lipschitz-continuity) at 0.\n\n## Equivalence of boundedness and continuity\nA linear operator between normed spaces is bounded if and only if it is [continuous](https://bliptext.com/articles/continuous-linear-operator).\n\n# In topological vector spaces\nA linear operator F : X \\to Y between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) is called a **' or just **' if whenever is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) in X then F(B) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) in Y. A subset of a TVS is called [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) (or more precisely, von Neumann [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)) if every neighborhood of the origin [absorbs](https://bliptext.com/articles/absorbing-set) it.\nIn a normed space (and even in a [seminormed space](https://bliptext.com/articles/seminormed-space)), a subset is von Neumann bounded if and only if it is norm bounded.\nHence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.\n\n## Continuity and boundedness\nEvery [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) linear operator between TVS is a bounded operator.\nThis implies that every continuous linear operator between metrizable TVS is bounded.\nHowever, in general, a bounded linear operator between two TVSs need not be continuous.\nThis formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets.\nIn this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.\nThis also means that boundedness is no longer equivalent to Lipschitz continuity in this context.\nIf the domain is a [bornological space](https://bliptext.com/articles/bornological-space) (for example, a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space), a [Fréchet space](https://bliptext.com/articles/fr-chet-space), a [normed space](https://bliptext.com/articles/normed-space)) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous.\nFor [LF spaces](https://bliptext.com/articles/lf-space), a weaker converse holds; any bounded linear map from an LF space is [sequentially continuous](https://bliptext.com/articles/sequentially-continuous).\nIf F : X \\to Y is a linear operator between two topological vector spaces and if there exists a neighborhood U of the origin in X such that F(U) is a bounded subset of Y, then F is continuous.\nThis fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous.\nIn particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a [normed space](https://bliptext.com/articles/normed-space)).\n\n### Bornological spaces\nBornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous.\nThat is, a locally convex TVS X is a bornological space if and only if for every locally convex TVS Y, a linear operator F : X \\to Y is continuous if and only if it is bounded.\nEvery normed space is bornological.\n\n## Characterizations of bounded linear operators\nLet F : X \\to Y be a linear operator between topological vector spaces (not necessarily Hausdorff).\nThe following are equivalent:\nif X and Y are [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then the following may be add to this list:\n

F maps bounded [disks](https://bliptext.com/articles/absolutely-convex-set) into bounded [disks](https://bliptext.com/articles/absolutely-convex-set).

F^{-1} maps [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in Y into [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in X.

\n\nif X is a [bornological space](https://bliptext.com/articles/bornological-space) and Y is locally convex then the following may be added to this list:\n

F is [sequentially continuous at some](https://bliptext.com/articles/sequential-continuity-at-a-point) (or equivalently, at every) point of its domain.\n

F is [sequentially continuous at the origin](https://bliptext.com/articles/sequential-continuity-at-a-point).

\n\n\n# Examples\n

Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed [matrix](https://bliptext.com/articles/matrix-mathematics).
Any linear operator defined on a finite-dimensional normed space is bounded.
On the [sequence space](https://bliptext.com/articles/sequence-space) c_{00} of eventually zero sequences of real numbers, considered with the \\ell^1 norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1.\nIf the same space is considered with the norm, the same operator is not bounded.
Many [integral transforms](https://bliptext.com/articles/integral-transform) are bounded linear operators.\nFor instance, if\nis a continuous function, then the operator L defined on the space C[a, b] of continuous functions on [a, b] endowed with the [uniform norm](https://bliptext.com/articles/uniform-norm) and with values in the space C[c, d] with L given by the formula\nis bounded.\nThis operator is in fact a [compact operator](https://bliptext.com/articles/compact-operator).\nThe compact operators form an important class of bounded operators.
The [Laplace operator](https://bliptext.com/articles/laplace-operator)\n(its [domain](https://bliptext.com/articles/domain-of-a-function) is a [Sobolev space](https://bliptext.com/articles/sobolev-space) and it takes values in a space of [square-integrable functions](https://bliptext.com/articles/square-integrable-function)) is bounded.
The [shift operator](https://bliptext.com/articles/shift-operator) on the [Lp space](https://bliptext.com/articles/lp-space) \\ell^2 of all [sequences](https://bliptext.com/articles/sequence) of real numbers with\nis bounded.\nIts operator norm is easily seen to be 1.

\n\n## Unbounded linear operators\nLet X be the space of all [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomial) on with the norm\nThe operator L : X \\to X that maps a polynomial to its [derivative](https://bliptext.com/articles/derivative) is not bounded.\nIndeed, for with we have while as so L is not bounded.\n\n## Properties of the space of bounded linear operators\nThe space of all bounded linear operators from X to Y is denoted by B(X, Y).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:41:07.469Z","updated_at":"2025-01-25T19:41:07.469Z","search_vector":"'bound':1 'oper':2","search_content":" \n \n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory), a **[bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) linear operator** is a [linear transformation](https://bliptext.com/articles/linear-transformation) L : X \\to Y between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) X and Y that maps [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subsets of X to [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subsets of Y. If X and Y are [normed vector spaces](https://bliptext.com/articles/normed-vector-space) (a special type of TVS), then L is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) if and only if there exists some M > 0 such that for all x \\in X,\nThe smallest such M is called the [operator norm](https://bliptext.com/articles/operator-norm) of L and denoted by \\|L\\|.\nA bounded operator between normed spaces is [continuous](https://bliptext.com/articles/continuous-linear-operator) and vice versa.\nThe concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.\nOutside of functional analysis, when a function f : X \\to Y is called \"[bounded](https://bliptext.com/articles/bounded-function)\" then this usually means that its [image](https://bliptext.com/articles/image-of-a-function) f(X) is a [bounded](https://bliptext.com/articles/bounded-function) subset of its codomain.\nA linear map has this property if and only if it is identically 0.\nConsequently, in functional analysis, when a linear operator is called \"bounded\" then it is never meant in this abstract sense (of having a bounded image).\n\n# In normed vector spaces\nEvery bounded operator is [Lipschitz continuous](https://bliptext.com/articles/lipschitz-continuity) at 0.\n\n## Equivalence of boundedness and continuity\nA linear operator between normed spaces is bounded if and only if it is [continuous](https://bliptext.com/articles/continuous-linear-operator).\n\n# In topological vector spaces\nA linear operator F : X \\to Y between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) is called a **' or just **' if whenever is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) in X then F(B) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) in Y. A subset of a TVS is called [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) (or more precisely, von Neumann [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)) if every neighborhood of the origin [absorbs](https://bliptext.com/articles/absorbing-set) it.\nIn a normed space (and even in a [seminormed space](https://bliptext.com/articles/seminormed-space)), a subset is von Neumann bounded if and only if it is norm bounded.\nHence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.\n\n## Continuity and boundedness\nEvery [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) linear operator between TVS is a bounded operator.\nThis implies that every continuous linear operator between metrizable TVS is bounded.\nHowever, in general, a bounded linear operator between two TVSs need not be continuous.\nThis formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets.\nIn this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.\nThis also means that boundedness is no longer equivalent to Lipschitz continuity in this context.\nIf the domain is a [bornological space](https://bliptext.com/articles/bornological-space) (for example, a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space), a [Fréchet space](https://bliptext.com/articles/fr-chet-space), a [normed space](https://bliptext.com/articles/normed-space)) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous.\nFor [LF spaces](https://bliptext.com/articles/lf-space), a weaker converse holds; any bounded linear map from an LF space is [sequentially continuous](https://bliptext.com/articles/sequentially-continuous).\nIf F : X \\to Y is a linear operator between two topological vector spaces and if there exists a neighborhood U of the origin in X such that F(U) is a bounded subset of Y, then F is continuous.\nThis fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous.\nIn particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a [normed space](https://bliptext.com/articles/normed-space)).\n\n### Bornological spaces\nBornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous.\nThat is, a locally convex TVS X is a bornological space if and only if for every locally convex TVS Y, a linear operator F : X \\to Y is continuous if and only if it is bounded.\nEvery normed space is bornological.\n\n## Characterizations of bounded linear operators\nLet F : X \\to Y be a linear operator between topological vector spaces (not necessarily Hausdorff).\nThe following are equivalent:\nif X and Y are [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then the following may be add to this list:\n F maps bounded [disks](https://bliptext.com/articles/absolutely-convex-set) into bounded [disks](https://bliptext.com/articles/absolutely-convex-set). \n F^{-1} maps [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in Y into [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in X. \n \nif X is a [bornological space](https://bliptext.com/articles/bornological-space) and Y is locally convex then the following may be added to this list:\n F is [sequentially continuous at some](https://bliptext.com/articles/sequential-continuity-at-a-point) (or equivalently, at every) point of its domain.\n F is [sequentially continuous at the origin](https://bliptext.com/articles/sequential-continuity-at-a-point). \n \n\n# Examples\n \n Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed [matrix](https://bliptext.com/articles/matrix-mathematics). \n Any linear operator defined on a finite-dimensional normed space is bounded. \n On the [sequence space](https://bliptext.com/articles/sequence-space) c_{00} of eventually zero sequences of real numbers, considered with the \\ell^1 norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1.\nIf the same space is considered with the norm, the same operator is not bounded. \n Many [integral transforms](https://bliptext.com/articles/integral-transform) are bounded linear operators.\nFor instance, if\nis a continuous function, then the operator L defined on the space C[a, b] of continuous functions on [a, b] endowed with the [uniform norm](https://bliptext.com/articles/uniform-norm) and with values in the space C[c, d] with L given by the formula\nis bounded.\nThis operator is in fact a [compact operator](https://bliptext.com/articles/compact-operator).\nThe compact operators form an important class of bounded operators. \n The [Laplace operator](https://bliptext.com/articles/laplace-operator)\n(its [domain](https://bliptext.com/articles/domain-of-a-function) is a [Sobolev space](https://bliptext.com/articles/sobolev-space) and it takes values in a space of [square-integrable functions](https://bliptext.com/articles/square-integrable-function)) is bounded. \n The [shift operator](https://bliptext.com/articles/shift-operator) on the [Lp space](https://bliptext.com/articles/lp-space) \\ell^2 of all [sequences](https://bliptext.com/articles/sequence) of real numbers with\nis bounded.\nIts operator norm is easily seen to be 1. \n \n\n## Unbounded linear operators\nLet X be the space of all [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomial) on with the norm\nThe operator L : X \\to X that maps a polynomial to its [derivative](https://bliptext.com/articles/derivative) is not bounded.\nIndeed, for with we have while as so L is not bounded.\n\n## Properties of the space of bounded linear operators\nThe space of all bounded linear operators from X to Y is denoted by B(X, Y).","revisionCount":0},"hasMore":false,"nextCursor":null}

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