{"history":[],"article":{"id":"25b9f0ce-6abe-4ad8-ba09-fa425b3d1c78","title":"Continuous linear operator","slug":"continuous-linear-operator","content":":::summary\n:::\n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a **[continuous](https://bliptext.com/articles/continuous-function-topology) linear operator** or **[continuous](https://bliptext.com/articles/continuous-function-topology) linear mapping** is a [continuous](https://bliptext.com/articles/continuous-function-topology) [linear transformation](https://bliptext.com/articles/linear-transformation) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nAn operator between two [normed spaces](https://bliptext.com/articles/normed-space) is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) if and only if it is a continuous linear operator.\n\n# Continuous linear operators\n\n## Characterizations of continuity\nSuppose that F : X \\to Y is a [linear operator](https://bliptext.com/articles/linear-operator) between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs).\nThe following are equivalent:\n

F is continuous.
F is [continuous at some point](https://bliptext.com/articles/continuity-at-a-point) x \\in X.
F is continuous at the origin in X.

\nIf Y is [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then this list may be extended to include:\n

for every continuous [seminorm](https://bliptext.com/articles/seminorm) q on Y, there exists a continuous [seminorm](https://bliptext.com/articles/seminorm) p on X such that

\n\nIf X and Y are both [Hausdorff](https://bliptext.com/articles/hausdorff-space) locally convex spaces then this list may be extended to include:\n

F is [weakly continuous](https://bliptext.com/articles/weakly-continuous) and its [transpose](https://bliptext.com/articles/transpose) maps [equicontinuous](https://bliptext.com/articles/equicontinuity) subsets of Y^{\\prime} to [equicontinuous](https://bliptext.com/articles/equicontinuity) subsets of X^{\\prime}.

\n\nIf X is a [sequential space](https://bliptext.com/articles/sequential-space) (such as a [pseudometrizable space](https://bliptext.com/articles/metrizable-topological-vector-space)) then this list may be extended to include:\n

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

\n\nIf X is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) or metrizable (such as a normed or [Banach space](https://bliptext.com/articles/banach-space)) then we may add to this list:\n

F is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) (that is, it maps bounded subsets of X to bounded subsets of Y).

\n\nIf Y is [seminormable space](https://bliptext.com/articles/seminormable-space) (such as a [normed space](https://bliptext.com/articles/normed-space)) then this list may be extended to include:\n

F maps some neighborhood of 0 to a bounded subset of Y.

\n\nIf X and Y are both [normed](https://bliptext.com/articles/normed-space) or semi[normed](https://bliptext.com/articles/normed-space) spaces (with both seminorms denoted by \\|\\cdot\\|) then this list may be extended to include:\n

for every r > 0 there exists some \\delta > 0 such that

\n\nIf X and Y are Hausdorff locally convex spaces with Y finite-dimensional then this list may be extended to include:\n

the graph of F is closed in X \\times Y.

\n\n\n# Continuity and boundedness\nThroughout, F : X \\to Y is a [linear map](https://bliptext.com/articles/linear-map) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs).\n**Bounded subset**\nThe notion of a \"bounded set\" for a topological vector space is that of being a [von Neumann bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space).\nIf the space happens to also be a [normed space](https://bliptext.com/articles/normed-space) (or a semi[normed space](https://bliptext.com/articles/normed-space)) then a subset S is von Neumann bounded if and only if it is, meaning that\nA subset of a normed (or seminormed) space is called if it is norm-bounded (or equivalently, von Neumann bounded).\nFor example, the scalar field (\\Reals or \\Complex) with the [absolute value](https://bliptext.com/articles/absolute-value) |\\cdot| is a normed space, so a subset S is bounded if and only if is finite, which happens if and only if S is contained in some open (or closed) ball centered at the origin (zero).\nAny translation, scalar multiple, and subset of a bounded set is again bounded.\n**Function bounded on a set**\nIf is a set then F : X \\to Y is said to be if F(S) is a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of Y, which if is a normed (or seminormed) space happens if and only if A linear map F is bounded on a set S if and only if it is bounded on for every x \\in X (because and any translation of a bounded set is again bounded) if and only if it is bounded on for every non-zero scalar c \\neq 0 (because and any scalar multiple of a bounded set is again bounded).\nConsequently, if is a normed or seminormed space, then a linear map F : X \\to Y is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin\n**Bounded linear maps**\nBy definition, a linear map F : X \\to Y between [TVSs](https://bliptext.com/articles/topological-vector-space) is said to be and is called a if for every [(von Neumann) bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of its domain, F(B) is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain.\nWhen the domain X is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin.\nExplicitly, if B_1 denotes this ball then F : X \\to Y is a bounded linear operator if and only if is a bounded subset of Y; if Y is also a (semi)normed space then this happens if and only if the [operator norm](https://bliptext.com/articles/operator-norm) is finite.\nEvery [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) linear operator is bounded.\n**Function bounded on a neighborhood and local boundedness**\nIn contrast, a map F : X \\to Y is said to be a point x \\in X or x if there exists a [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) U of this point in X such that F(U) is a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of Y. It is \"\" (of some point) if there exists point x in its domain at which it is locally bounded, in which case this linear map F is necessarily locally bounded at point of its domain.\nThe term \"[Locally bounded function](https://bliptext.com/articles/locally-bounded-function)\" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define \"locally bounded\" to instead be a synonym of \"[bounded linear operator](https://bliptext.com/articles/bounded-linear-operator)\", which are related but equivalent concepts.\nFor this reason, this article will avoid the term \"locally bounded\" and instead say \"locally bounded at every point\" (there is no disagreement about the definition of \"locally bounded \").\n\n## Bounded on a neighborhood implies continuous implies bounded\nA linear map is \"[bounded on a neighborhood](https://bliptext.com/articles/)\" (of some point) if and only if it is locally [bounded](https://bliptext.com/articles/bounded-linear-operator) at every point of its domain, in which case it is necessarily [continuous](https://bliptext.com/articles/continuous-function-topology) (even if its domain is not a [normed space](https://bliptext.com/articles/normed-space)) and thus also [bounded](https://bliptext.com/articles/bounded-linear-operator) (because a [continuous](https://bliptext.com/articles/continuous-function-topology) linear operator is always a [bounded](https://bliptext.com/articles/bounded-linear-operator) linear operator).\nFor any linear map, if it is [bounded on a neighborhood](https://bliptext.com/articles/) then it is continuous, and if it is continuous then it is [bounded](https://bliptext.com/articles/bounded-linear-operator).\nThe converse statements are not true in general but they are both true when the linear map's domain is a [normed space](https://bliptext.com/articles/normed-space).\nExamples and additional details are now given below.\n\n### Continuous and bounded but not bounded on a neighborhood\nThe next example shows that it is possible for a linear map to be [continuous](https://bliptext.com/articles/continuous-function-topology) (and thus also bounded) but not bounded on any neighborhood.\nIn particular, it demonstrates that being \"[bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood\" is always synonymous with being \"[bounded](https://bliptext.com/articles/bounded-linear-operator)\".\nThis shows that it is possible for a linear map to be continuous but bounded on any neighborhood.\nIndeed, this example shows that every [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) that is not seminormable has a linear TVS-[automorphism](https://bliptext.com/articles/automorphism) that is not bounded on any neighborhood of any point.\nThus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.\n\n## Guaranteeing converses\nTo summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being [bounded](https://bliptext.com/articles/bounded-linear-operator), and being [bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood are all [equivalent](https://bliptext.com/articles/logical-equivalence).\nA linear map whose domain codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.\nAnd a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) valued in a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) will be continuous if its domain is [(pseudo)metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) or [bornological](https://bliptext.com/articles/bornological-space).\n**Guaranteeing that \"continuous\" implies \"bounded on a neighborhood\"**\nA TVS is said to be if there exists a neighborhood that is also a [bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space).\nFor example, every [normed](https://bliptext.com/articles/normed-space) or semi[normed](https://bliptext.com/articles/normed-space) space is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.\nIf B is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood B).\nConsequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is [bounded on a neighborhood](https://bliptext.com/articles/).\nMoreover, any TVS with this property must be a locally bounded TVS.\nExplicitly, if X is a TVS such that every continuous linear map (into any TVS) whose domain is X is necessarily bounded on a neighborhood, then X must be a locally bounded TVS (because the [identity function](https://bliptext.com/articles/identity-function) X \\to X is always a continuous linear map).\nAny linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.\nConversely, if Y is a TVS such that every continuous linear map (from any TVS) with codomain Y is necessarily [bounded on a neighborhood](https://bliptext.com/articles/), then Y must be a locally bounded TVS.\nIn particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.\nThus when the domain the codomain of a linear map is normable or seminormable, then continuity will be [equivalent](https://bliptext.com/articles/logical-equivalence) to being bounded on a neighborhood.\n**Guaranteeing that \"bounded\" implies \"continuous\"**\nA continuous linear operator is always a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator).\nBut importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be [bounded](https://bliptext.com/articles/bounded-linear-operator) but to be continuous.\nA linear map whose domain is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) (such as any [normed space](https://bliptext.com/articles/normed-space)) is [bounded](https://bliptext.com/articles/bounded-linear-operator) if and only if it is continuous.\nThe same is true of a linear map from a [bornological space](https://bliptext.com/articles/bornological-space) into a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space).\n**Guaranteeing that \"bounded\" implies \"bounded on a neighborhood\"**\nIn general, without additional information about either the linear map or its domain or codomain, the map being \"bounded\" is not equivalent to it being \"bounded on a neighborhood\".\nIf F : X \\to Y is a bounded linear operator from a [normed space](https://bliptext.com/articles/normed-space) X into some TVS then F : X \\to Y is necessarily continuous; this is because any open ball B centered at the origin in X is both a bounded subset (which implies that F(B) is bounded since F is a bounded linear map) and a neighborhood of the origin in X, so that F is thus bounded on this neighborhood B of the origin, which (as mentioned above) guarantees continuity.\n\n# Continuous linear functionals\nEvery linear functional on a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them.\nHowever, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.\n\n## Characterizing continuous linear functionals\nLet X be a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} (X need not be [Hausdorff](https://bliptext.com/articles/hausdorff-space) or [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space)) and let be a [linear functional](https://bliptext.com/articles/linear-functional) on X. The following are equivalent:\n

f is continuous.
f is uniformly continuous on X.
f is [continuous at some point](https://bliptext.com/articles/continuity-at-a-point) of X.\n
f is continuous at the origin.\n
f is [bounded on a neighborhood](https://bliptext.com/articles/) (of some point).\nSaid differently, f is a [locally bounded at some point](https://bliptext.com/articles/) of its domain.\n
f is [bounded on a neighborhood of the origin](https://bliptext.com/articles/).\nSaid differently, f is a [locally bounded at the origin.](https://bliptext.com/articles/)\n
There exists some neighborhood U of the origin such that\n
f is a [locally bounded at every point](https://bliptext.com/articles/) of its domain.
The kernel of f is closed in X.
Either f = 0 or else the kernel of f is dense in X.
There exists a continuous seminorm p on X such that |f| \\leq p.
The graph of f is closed.
is continuous, where denotes the [real part](https://bliptext.com/articles/real-part) of f.

\nIf X and Y are complex vector spaces then this list may be extended to include:\n

The imaginary part of f is continuous.

\n\nIf the domain X is a [sequential space](https://bliptext.com/articles/sequential-space) then this list may be extended to include:\n

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

\n\nIf the domain X is [metrizable or pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) (for example, a [Fréchet space](https://bliptext.com/articles/fr-chet-space) or a [normed space](https://bliptext.com/articles/normed-space)) then this list may be extended to include:\n

f is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) (that is, it maps bounded subsets of its domain to bounded subsets of its codomain).

\n\nIf the domain X is a [bornological space](https://bliptext.com/articles/bornological-space) (for example, a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space)) and Y is [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then this list may be extended to include:\n

f is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator).

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

f is sequentially continuous at the origin.

\n\nand if in addition X is a vector space over the [real numbers](https://bliptext.com/articles/real-numbers) (which in particular, implies that f is real-valued) then this list may be extended to include:\n

There exists a continuous seminorm p on X such that f \\leq p.

For some real r, the half-space is closed.

For any real r, the half-space is closed.

\n\nIf X is complex then either all three of f, and are [continuous](https://bliptext.com/articles/continuous-linear-map) (respectively, [bounded](https://bliptext.com/articles/bounded-linear-operator)), or else all three are dis[continuous](https://bliptext.com/articles/continuous-linear-map) (respectively, un[bounded](https://bliptext.com/articles/bounded-linear-operator)).\n\n# Examples\nEvery linear map whose domain is a finite-dimensional Hausdorff [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is continuous.\nThis is not true if the finite-dimensional TVS is not Hausdorff.\nEvery (constant) map X \\to Y between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood X of the origin.\nIn particular, every TVS has a non-empty [continuous dual space](https://bliptext.com/articles/continuous-dual-space) (although it is possible for the constant zero map to be its only continuous linear functional).\nSuppose X is any Hausdorff TVS.\nThen [linear functional](https://bliptext.com/articles/linear-functional) on X is necessarily continuous if and only if every vector subspace of X is closed.\nEvery linear functional on X is necessarily a bounded linear functional if and only if every [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of X is contained in a finite-dimensional vector subspace.\n\n# Properties\nA [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space) is [normable](https://bliptext.com/articles/normable) if and only if every bounded linear functional on it is continuous.\nA continuous linear operator maps [bounded sets](https://bliptext.com/articles/bounded-set-topological-vector-space) into [bounded sets](https://bliptext.com/articles/bounded-set-topological-vector-space).\nThe proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality\nfor any subset D of Y and any x \\in X, which is true due to the [additivity](https://bliptext.com/articles/additive-map) of F.\n\n## Properties of continuous linear functionals\nIf X is a complex [normed space](https://bliptext.com/articles/normed-space) and f is a linear functional on X, then (where in particular, one side is infinite if and only if the other side is infinite).\nEvery non-trivial continuous linear functional on a TVS X is an [open map](https://bliptext.com/articles/open-map).\nIf f is a linear functional on a real vector space X and if p is a seminorm on X, then |f| \\leq p if and only if f \\leq p.\nIf is a linear functional and is a non-empty subset, then by defining the sets\nthe supremum can be written more succinctly as because\nIf s is a scalar then\nso that if r > 0 is a real number and is the closed ball of radius r centered at the origin then the following are equivalent:","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:39:03.164Z","updated_at":"2025-01-25T21:39:03.164Z","search_vector":"'continu':1 'linear':2 'oper':3","search_content":" \n \n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a **[continuous](https://bliptext.com/articles/continuous-function-topology) linear operator** or **[continuous](https://bliptext.com/articles/continuous-function-topology) linear mapping** is a [continuous](https://bliptext.com/articles/continuous-function-topology) [linear transformation](https://bliptext.com/articles/linear-transformation) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nAn operator between two [normed spaces](https://bliptext.com/articles/normed-space) is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) if and only if it is a continuous linear operator.\n\n# Continuous linear operators\n\n## Characterizations of continuity\nSuppose that F : X \\to Y is a [linear operator](https://bliptext.com/articles/linear-operator) between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs).\nThe following are equivalent:\n \n F is continuous. \n F is [continuous at some point](https://bliptext.com/articles/continuity-at-a-point) x \\in X. \n F is continuous at the origin in X. \n \nIf Y is [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then this list may be extended to include:\n for every continuous [seminorm](https://bliptext.com/articles/seminorm) q on Y, there exists a continuous [seminorm](https://bliptext.com/articles/seminorm) p on X such that \n \nIf X and Y are both [Hausdorff](https://bliptext.com/articles/hausdorff-space) locally convex spaces then this list may be extended to include:\n F is [weakly continuous](https://bliptext.com/articles/weakly-continuous) and its [transpose](https://bliptext.com/articles/transpose) maps [equicontinuous](https://bliptext.com/articles/equicontinuity) subsets of Y^{\\prime} to [equicontinuous](https://bliptext.com/articles/equicontinuity) subsets of X^{\\prime}. \n \nIf X is a [sequential space](https://bliptext.com/articles/sequential-space) (such as a [pseudometrizable space](https://bliptext.com/articles/metrizable-topological-vector-space)) then this list may be extended to include:\n F is [sequentially continuous](https://bliptext.com/articles/sequential-continuity-at-a-point) at some (or equivalently, at every) point of its domain. \n \nIf X is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) or metrizable (such as a normed or [Banach space](https://bliptext.com/articles/banach-space)) then we may add to this list:\n F is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) (that is, it maps bounded subsets of X to bounded subsets of Y). \n \nIf Y is [seminormable space](https://bliptext.com/articles/seminormable-space) (such as a [normed space](https://bliptext.com/articles/normed-space)) then this list may be extended to include:\n F maps some neighborhood of 0 to a bounded subset of Y. \n \nIf X and Y are both [normed](https://bliptext.com/articles/normed-space) or semi[normed](https://bliptext.com/articles/normed-space) spaces (with both seminorms denoted by \\|\\cdot\\|) then this list may be extended to include:\n for every r > 0 there exists some \\delta > 0 such that \n \nIf X and Y are Hausdorff locally convex spaces with Y finite-dimensional then this list may be extended to include:\n the graph of F is closed in X \\times Y. \n \n\n# Continuity and boundedness\nThroughout, F : X \\to Y is a [linear map](https://bliptext.com/articles/linear-map) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs).\n**Bounded subset**\nThe notion of a \"bounded set\" for a topological vector space is that of being a [von Neumann bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space).\nIf the space happens to also be a [normed space](https://bliptext.com/articles/normed-space) (or a semi[normed space](https://bliptext.com/articles/normed-space)) then a subset S is von Neumann bounded if and only if it is, meaning that\nA subset of a normed (or seminormed) space is called if it is norm-bounded (or equivalently, von Neumann bounded).\nFor example, the scalar field (\\Reals or \\Complex) with the [absolute value](https://bliptext.com/articles/absolute-value) |\\cdot| is a normed space, so a subset S is bounded if and only if is finite, which happens if and only if S is contained in some open (or closed) ball centered at the origin (zero).\nAny translation, scalar multiple, and subset of a bounded set is again bounded.\n**Function bounded on a set**\nIf is a set then F : X \\to Y is said to be if F(S) is a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of Y, which if is a normed (or seminormed) space happens if and only if A linear map F is bounded on a set S if and only if it is bounded on for every x \\in X (because and any translation of a bounded set is again bounded) if and only if it is bounded on for every non-zero scalar c \\neq 0 (because and any scalar multiple of a bounded set is again bounded).\nConsequently, if is a normed or seminormed space, then a linear map F : X \\to Y is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin\n**Bounded linear maps**\nBy definition, a linear map F : X \\to Y between [TVSs](https://bliptext.com/articles/topological-vector-space) is said to be and is called a if for every [(von Neumann) bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of its domain, F(B) is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain.\nWhen the domain X is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin.\nExplicitly, if B_1 denotes this ball then F : X \\to Y is a bounded linear operator if and only if is a bounded subset of Y; if Y is also a (semi)normed space then this happens if and only if the [operator norm](https://bliptext.com/articles/operator-norm) is finite.\nEvery [sequentially continuous](https://bliptext.com/articles/sequentially-continuous) linear operator is bounded.\n**Function bounded on a neighborhood and local boundedness**\nIn contrast, a map F : X \\to Y is said to be a point x \\in X or x if there exists a [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) U of this point in X such that F(U) is a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of Y. It is \"\" (of some point) if there exists point x in its domain at which it is locally bounded, in which case this linear map F is necessarily locally bounded at point of its domain.\nThe term \"[Locally bounded function](https://bliptext.com/articles/locally-bounded-function)\" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define \"locally bounded\" to instead be a synonym of \"[bounded linear operator](https://bliptext.com/articles/bounded-linear-operator)\", which are related but equivalent concepts.\nFor this reason, this article will avoid the term \"locally bounded\" and instead say \"locally bounded at every point\" (there is no disagreement about the definition of \"locally bounded \").\n\n## Bounded on a neighborhood implies continuous implies bounded\nA linear map is \"[[bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood](https://bliptext.com/articles/)\" (of some point) if and only if it is locally [bounded](https://bliptext.com/articles/bounded-linear-operator) at every point of its domain, in which case it is necessarily [continuous](https://bliptext.com/articles/continuous-function-topology) (even if its domain is not a [normed space](https://bliptext.com/articles/normed-space)) and thus also [bounded](https://bliptext.com/articles/bounded-linear-operator) (because a [continuous](https://bliptext.com/articles/continuous-function-topology) linear operator is always a [bounded](https://bliptext.com/articles/bounded-linear-operator) linear operator).\nFor any linear map, if it is [[bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood](https://bliptext.com/articles/) then it is continuous, and if it is continuous then it is [bounded](https://bliptext.com/articles/bounded-linear-operator).\nThe converse statements are not true in general but they are both true when the linear map's domain is a [normed space](https://bliptext.com/articles/normed-space).\nExamples and additional details are now given below.\n\n### Continuous and bounded but not bounded on a neighborhood\nThe next example shows that it is possible for a linear map to be [continuous](https://bliptext.com/articles/continuous-function-topology) (and thus also bounded) but not bounded on any neighborhood.\nIn particular, it demonstrates that being \"[bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood\" is always synonymous with being \"[bounded](https://bliptext.com/articles/bounded-linear-operator)\".\nThis shows that it is possible for a linear map to be continuous but bounded on any neighborhood.\nIndeed, this example shows that every [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) that is not seminormable has a linear TVS-[automorphism](https://bliptext.com/articles/automorphism) that is not bounded on any neighborhood of any point.\nThus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.\n\n## Guaranteeing converses\nTo summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being [bounded](https://bliptext.com/articles/bounded-linear-operator), and being [bounded](https://bliptext.com/articles/bounded-linear-operator) on a neighborhood are all [equivalent](https://bliptext.com/articles/logical-equivalence).\nA linear map whose domain codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.\nAnd a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) valued in a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) will be continuous if its domain is [(pseudo)metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) or [bornological](https://bliptext.com/articles/bornological-space).\n**Guaranteeing that \"continuous\" implies \"bounded on a neighborhood\"**\nA TVS is said to be if there exists a neighborhood that is also a [bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space).\nFor example, every [normed](https://bliptext.com/articles/normed-space) or semi[normed](https://bliptext.com/articles/normed-space) space is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.\nIf B is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood B).\nConsequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is [bounded on a neighborhood](https://bliptext.com/articles/).\nMoreover, any TVS with this property must be a locally bounded TVS.\nExplicitly, if X is a TVS such that every continuous linear map (into any TVS) whose domain is X is necessarily bounded on a neighborhood, then X must be a locally bounded TVS (because the [identity function](https://bliptext.com/articles/identity-function) X \\to X is always a continuous linear map).\nAny linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.\nConversely, if Y is a TVS such that every continuous linear map (from any TVS) with codomain Y is necessarily [bounded on a neighborhood](https://bliptext.com/articles/), then Y must be a locally bounded TVS.\nIn particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.\nThus when the domain the codomain of a linear map is normable or seminormable, then continuity will be [equivalent](https://bliptext.com/articles/logical-equivalence) to being bounded on a neighborhood.\n**Guaranteeing that \"bounded\" implies \"continuous\"**\nA continuous linear operator is always a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator).\nBut importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be [bounded](https://bliptext.com/articles/bounded-linear-operator) but to be continuous.\nA linear map whose domain is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) (such as any [normed space](https://bliptext.com/articles/normed-space)) is [bounded](https://bliptext.com/articles/bounded-linear-operator) if and only if it is continuous.\nThe same is true of a linear map from a [bornological space](https://bliptext.com/articles/bornological-space) into a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space).\n**Guaranteeing that \"bounded\" implies \"bounded on a neighborhood\"**\nIn general, without additional information about either the linear map or its domain or codomain, the map being \"bounded\" is not equivalent to it being \"bounded on a neighborhood\".\nIf F : X \\to Y is a bounded linear operator from a [normed space](https://bliptext.com/articles/normed-space) X into some TVS then F : X \\to Y is necessarily continuous; this is because any open ball B centered at the origin in X is both a bounded subset (which implies that F(B) is bounded since F is a bounded linear map) and a neighborhood of the origin in X, so that F is thus bounded on this neighborhood B of the origin, which (as mentioned above) guarantees continuity.\n\n# Continuous linear functionals\nEvery linear functional on a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them.\nHowever, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.\n\n## Characterizing continuous linear functionals\nLet X be a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} (X need not be [Hausdorff](https://bliptext.com/articles/hausdorff-space) or [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space)) and let be a [linear functional](https://bliptext.com/articles/linear-functional) on X. The following are equivalent:\n \n f is continuous. \n f is uniformly continuous on X. \n f is [continuous at some point](https://bliptext.com/articles/continuity-at-a-point) of X.\n \n f is continuous at the origin.\n \n f is [bounded on a neighborhood](https://bliptext.com/articles/) (of some point).\nSaid differently, f is a [locally bounded at some point](https://bliptext.com/articles/) of its domain.\n \n f is [bounded on a neighborhood of the origin](https://bliptext.com/articles/).\nSaid differently, f is a [locally bounded at the origin.](https://bliptext.com/articles/)\n \n There exists some neighborhood U of the origin such that\n \n f is a [locally bounded at every point](https://bliptext.com/articles/) of its domain. \n The kernel of f is closed in X. \n Either f = 0 or else the kernel of f is dense in X. \n There exists a continuous seminorm p on X such that |f| \\leq p. The graph of f is closed. \n is continuous, where denotes the [real part](https://bliptext.com/articles/real-part) of f. \n \nIf X and Y are complex vector spaces then this list may be extended to include:\n The imaginary part of f is continuous. \n \nIf the domain X is a [sequential space](https://bliptext.com/articles/sequential-space) then this list may be extended to include:\n f is [sequentially continuous](https://bliptext.com/articles/sequential-continuity-at-a-point) at some (or equivalently, at every) point of its domain. \n \nIf the domain X is [metrizable or pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) (for example, a [Fréchet space](https://bliptext.com/articles/fr-chet-space) or a [normed space](https://bliptext.com/articles/normed-space)) then this list may be extended to include:\n f is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) (that is, it maps bounded subsets of its domain to bounded subsets of its codomain). \n \nIf the domain X is a [bornological space](https://bliptext.com/articles/bornological-space) (for example, a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space)) and Y is [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) then this list may be extended to include:\n f is a [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator). \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. \n \nand if in addition X is a vector space over the [real numbers](https://bliptext.com/articles/real-numbers) (which in particular, implies that f is real-valued) then this list may be extended to include:\n There exists a continuous seminorm p on X such that f \\leq p. \n For some real r, the half-space is closed. \n For any real r, the half-space is closed. \n \nIf X is complex then either all three of f, and are [continuous](https://bliptext.com/articles/continuous-linear-map) (respectively, [bounded](https://bliptext.com/articles/bounded-linear-operator)), or else all three are dis[continuous](https://bliptext.com/articles/continuous-linear-map) (respectively, un[bounded](https://bliptext.com/articles/bounded-linear-operator)).\n\n# Examples\nEvery linear map whose domain is a finite-dimensional Hausdorff [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is continuous.\nThis is not true if the finite-dimensional TVS is not Hausdorff.\nEvery (constant) map X \\to Y between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood X of the origin.\nIn particular, every TVS has a non-empty [continuous dual space](https://bliptext.com/articles/continuous-dual-space) (although it is possible for the constant zero map to be its only continuous linear functional).\nSuppose X is any Hausdorff TVS.\nThen [linear functional](https://bliptext.com/articles/linear-functional) on X is necessarily continuous if and only if every vector subspace of X is closed.\nEvery linear functional on X is necessarily a bounded linear functional if and only if every [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of X is contained in a finite-dimensional vector subspace.\n\n# Properties\nA [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space) is [normable](https://bliptext.com/articles/normable) if and only if every bounded linear functional on it is continuous.\nA continuous linear operator maps [bounded sets](https://bliptext.com/articles/bounded-set-topological-vector-space) into [bounded sets](https://bliptext.com/articles/bounded-set-topological-vector-space).\nThe proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality\nfor any subset D of Y and any x \\in X, which is true due to the [additivity](https://bliptext.com/articles/additive-map) of F.\n\n## Properties of continuous linear functionals\nIf X is a complex [normed space](https://bliptext.com/articles/normed-space) and f is a linear functional on X, then (where in particular, one side is infinite if and only if the other side is infinite).\nEvery non-trivial continuous linear functional on a TVS X is an [open map](https://bliptext.com/articles/open-map).\nIf f is a linear functional on a real vector space X and if p is a seminorm on X, then |f| \\leq p if and only if f \\leq p.\nIf is a linear functional and is a non-empty subset, then by defining the sets\nthe supremum can be written more succinctly as because\nIf s is a scalar then\nso that if r > 0 is a real number and is the closed ball of radius r centered at the origin then the following are equivalent:","revisionCount":0},"hasMore":false,"nextCursor":null}

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