{"history":[],"article":{"id":"d979ec63-e718-40e4-a589-6fef4584bb00","title":"Bounded set (topological vector space)","slug":"bounded-set-topological-vector-space","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 [set](https://bliptext.com/articles/set-mathematics) in a [topological vector space](https://bliptext.com/articles/topological-vector-space) is called **bounded** or von Neumann **bounded**, if every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the [zero vector](https://bliptext.com/articles/zero-vector) can be inflated to include the [set](https://bliptext.com/articles/set-mathematics).\nA set that is not bounded is called **unbounded**.\nBounded sets are a natural way to define [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [polar topologies](https://bliptext.com/articles/polar-topology) on the [vector spaces](https://bliptext.com/articles/vector-space) in a [dual pair](https://bliptext.com/articles/dual-pair), as the [polar set](https://bliptext.com/articles/polar-set) of a bounded set is an [absolutely convex](https://bliptext.com/articles/absolutely-convex-set) and [absorbing set](https://bliptext.com/articles/absorbing-set).\nThe concept was first introduced by [John von Neumann](https://bliptext.com/articles/john-von-neumann) and [Andrey Kolmogorov](https://bliptext.com/articles/andrey-kolmogorov) in [1935](https://bliptext.com/articles/1935).\n\n# Definition\nSuppose X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over a [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K}.\nA subset B of X is called **' or just **' in X if any of the following equivalent conditions are satisfied:\n

: For every [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) V of the origin there exists a real r > 0 such that for all scalars s satisfying |s| \\geq r.
B is [absorbed](https://bliptext.com/articles/absorbing-set) by every [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of the origin.
For every neighborhood V of the origin there exists a scalar s such that
For every neighborhood V of the origin there exists a real r > 0 such that for all scalars s satisfying |s| \\leq r.
For every neighborhood V of the origin there exists a real r > 0 such that for all real
Any one of statements (1) through (5) above but with the word \"neighborhood\" replaced by any of the following: \"[balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"open [balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"closed [balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"open neighborhood,\" \"closed neighborhood\".\n
For every sequence of scalars that converges to 0 and every sequence in B, the sequence converges to 0 in X.
For every sequence in B, the sequence converges to 0 in X.
Every [countable](https://bliptext.com/articles/countable-set) subset of B is bounded (according to any defining condition other than this one).

\nIf \\mathcal{B} is a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) for X at the origin then this list may be extended to include:\n

Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to \nIf X is a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) space whose topology is defined by a family \\mathcal{P} of continuous [seminorms](https://bliptext.com/articles/seminorm), then this list may be extended to include:\n

p(B) is bounded for all

There exists a sequence of non-zero scalars such that for every sequence in B, the sequence is bounded in X (according to any defining condition other than this one).

For all B is bounded (according to any defining condition other than this one) in the [semi normed space](https://bliptext.com/articles/semi-normed-space) (X, p).

B is weakly bounded, i.e. every continuous linear functional is bounded on B

\n\nIf X is a [normed space](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-space) with [norm](https://bliptext.com/articles/norm-mathematics) \\|\\cdot\\| (or more generally, if it is a semi[normed space](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-space) and \\|\\cdot\\| is merely a semi[norm](https://bliptext.com/articles/norm-mathematics)), then this list may be extended to include:\n

B is a norm bounded subset of By definition, this means that there exists a real number r > 0 such that for all b \\in B.

B is a subset of some (open or closed) ball.\n\nIf B is a vector subspace of the TVS X then this list may be extended to include:\n

B is contained in the closure of \\{0\\}.\n\nA subset that is not bounded is called.\n\n## Bornology and fundamental systems of bounded sets\nThe collection of all bounded sets on a topological vector space X is called the or the\nA or of X is a set \\mathcal{B} of bounded subsets of X such that every bounded subset of X is a subset of some\nThe set of all bounded subsets of X trivially forms a fundamental system of bounded sets of X.\n\n### Examples\nIn any [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) TVS, the set of closed and bounded [disks](https://bliptext.com/articles/absolutely-convex-set) are a base of bounded set.\n\n# Examples and sufficient conditions\nUnless indicated otherwise, a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) need not be [Hausdorff](https://bliptext.com/articles/hausdorff-space) nor [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space).\n

Finite sets are bounded.
Every [totally bounded](https://bliptext.com/articles/totally-bounded) subset of a TVS is bounded.
Every [relatively compact set](https://bliptext.com/articles/relatively-compact-set) in a topological vector space is bounded.\nIf the space is equipped with the [weak topology](https://bliptext.com/articles/weak-topology-polar-topology) the converse is also true.
The set of points of a [Cauchy) sequence](https://bliptext.com/articles/cauchy-sequence) is bounded, the set of points of a [Cauchy](https://bliptext.com/articles/cauchy-[net](https://bliptext.com/articles/net-mathematics)) [net](https://bliptext.com/articles/net-mathematics) need not be bounded.
The closure of the origin (referring to the closure of the set \\{0\\}) is always a bounded closed vector subspace.\nThis set is the unique largest (with respect to set inclusion ) bounded vector subspace of X. In particular, if is a bounded subset of X then so is

\n**Unbounded sets**\nA set that is not bounded is said to be unbounded.\nAny vector subspace of a TVS that is not a contained in the closure of \\{0\\} is unbounded\nThere exists a [Fréchet space](https://bliptext.com/articles/fr-chet-space) X having a bounded subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any bounded subset of M.\n\n## Stability properties\n

In any TVS, finite [unions](https://bliptext.com/articles/union-set-theory), finite [Minkowski sums](https://bliptext.com/articles/minkowski-sum), scalar multiples, translations, subsets, [closures](https://bliptext.com/articles/closure-topology), [interiors](https://bliptext.com/articles/interior-topology), and [balanced hulls](https://bliptext.com/articles/balanced-set) of bounded sets are again bounded.
In any [locally convex TVS](https://bliptext.com/articles/locally-convex-topological-vector-space), the [convex hull](https://bliptext.com/articles/convex-set) (also called the [convex envelope](https://bliptext.com/articles/convex-envelope)) of a bounded set is again bounded.\nHowever, this may be false if the space is not locally convex, as the (non-locally convex) [Lp space](https://bliptext.com/articles/lp-space) L^p spaces for 0 < p < 1 have no nontrivial open convex subsets.
The image of a bounded set under a [continuous linear map](https://bliptext.com/articles/continuous-linear-map) is a bounded subset of the codomain.
A subset of an arbitrary [(Cartesian) product](https://bliptext.com/articles/product-space) of TVSs is bounded if and only if its image under every coordinate projections is bounded.
If and X is a topological vector subspace of Y, then S is bounded in X if and only if S is bounded in Y.

\n\n# Properties\nA [locally convex topological vector space](https://bliptext.com/articles/locally-convex-topological-vector-space) has a bounded neighborhood of zero [if and only if](https://bliptext.com/articles/if-and-only-if) its topology can be defined by a [seminorm](https://bliptext.com/articles/seminorm).\nThe [polar](https://bliptext.com/articles/polar-set) of a bounded set is an [absolutely convex](https://bliptext.com/articles/absolutely-convex-set) and [absorbing set](https://bliptext.com/articles/absorbing-set).\nUsing the definition of [uniformly bounded sets](https://bliptext.com/articles/) given below, [Mackey's countability condition](https://bliptext.com/articles/) can be restated as: If are bounded subsets of a [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) then there exists a sequence of positive real numbers such that are [uniformly bounded](https://bliptext.com/articles/).\nIn words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.\n\n# Generalizations\n\n## Uniformly bounded sets\nA [family of sets](https://bliptext.com/articles/family-of-sets) \\mathcal{B} of subsets of a [topological vector space](https://bliptext.com/articles/topological-vector-space) Y is said to be in Y, if there exists some bounded subset D of Y such that\nwhich happens if and only if its union\nis a bounded subset of Y. In the case of a [normed](https://bliptext.com/articles/normed-space) (or semi[normed](https://bliptext.com/articles/normed-space)) space, a family \\mathcal{B} is uniformly bounded if and only if its union is norm bounded, meaning that there exists some real M \\geq 0 such that for every or equivalently, if and only if\nA set H of maps from X to Y is said to be if the family is uniformly bounded in Y, which by definition means that there exists some bounded subset D of Y such that or equivalently, if and only if is a bounded subset of Y. A set H of linear maps between two normed (or seminormed) spaces X and Y is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in X if and only if their [operator norms](https://bliptext.com/articles/operator-norm) are uniformly bounded; that is, if and only if\nAssume H is equicontinuous and let W be a neighborhood of the origin in Y. Since H is equicontinuous, there exists a neighborhood U of the origin in X such that for every h \\in H. Because C is bounded in X, there exists some real r > 0 such that if t \\geq r then\nSo for every h \\in H and every t \\geq r, which implies that Thus is bounded in Y. [Q.E.D.](https://bliptext.com/articles/q-e-d)\nLet W be a [balanced](https://bliptext.com/articles/balanced-set) neighborhood of the origin in Y and let V be a closed [balanced](https://bliptext.com/articles/balanced-set) neighborhood of the origin in Y such that\nDefine\nwhich is a closed subset of X (since V is closed while every h : X \\to Y is continuous) that satisfies for every h \\in H. Note that for every non-zero scalar n \\neq 0, the set n E is closed in X (since scalar multiplication by n \\neq 0 is a [homeomorphism](https://bliptext.com/articles/homeomorphism)) and so every C \\cap n E is closed in C.\nIt will now be shown that from which follows.\nIf c \\in C then H(c) being bounded guarantees the existence of some positive integer such that where the linearity of every h \\in H now implies thus and hence as desired.\nThus\nexpresses C as a countable union of closed (in C) sets.\nSince C is a [nonmeager subset](https://bliptext.com/articles/nonmeager-subset) of itself (as it is a [Baire space](https://bliptext.com/articles/baire-space) by the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem)), this is only possible if there is some integer n \\in \\N such that C \\cap n E has non-empty interior in C. Let be any point belonging to this open subset of C. Let U be any balanced open neighborhood of the origin in X such that\nThe sets form an increasing (meaning p \\leq q implies ) cover of the compact space C, so there exists some p > 1 such that (and thus ).\nIt will be shown that for every h \\in H, thus demonstrating that is uniformly bounded in Y and completing the proof.\nSo fix h \\in H and c \\in C. Let\nThe convexity of C guarantees z \\in C and moreover, z \\in k + U since\nThus which is a subset of Since n V is balanced and we have which combined with gives\nFinally, and imply\nas desired.\n[Q.E.D.](https://bliptext.com/articles/q-e-d)\nSince every [singleton subset](https://bliptext.com/articles/singleton-set) of X is also a bounded subset, it follows that if is an [equicontinuous](https://bliptext.com/articles/equicontinuity) set of [continuous linear operators](https://bliptext.com/articles/continuous-linear-operator) between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) X and Y (not necessarily [Hausdorff](https://bliptext.com/articles/hausdorff-space) or locally convex), then the [orbit](https://bliptext.com/articles/orbit-group-theory) of every x \\in X is a bounded subset of Y.\n\n## Bounded subsets of topological modules\nThe definition of bounded sets can be generalized to [topological modules](https://bliptext.com/articles/topological-module).\nA subset A of a topological module M over a [topological ring](https://bliptext.com/articles/topological-ring) R is bounded if for any neighborhood N of 0_M there exists a neighborhood w of 0_R such that","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:37:57.582Z","updated_at":"2025-01-25T21:37:57.582Z","search_vector":"'bound':1 'set':2 'space':5 'topolog':3 'vector':4","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 [set](https://bliptext.com/articles/set-mathematics) in a [topological vector space](https://bliptext.com/articles/topological-vector-space) is called **bounded** or von Neumann **bounded**, if every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the [zero vector](https://bliptext.com/articles/zero-vector) can be inflated to include the [set](https://bliptext.com/articles/set-mathematics).\nA set that is not bounded is called **unbounded**.\nBounded sets are a natural way to define [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [polar topologies](https://bliptext.com/articles/polar-topology) on the [vector spaces](https://bliptext.com/articles/vector-space) in a [dual pair](https://bliptext.com/articles/dual-pair), as the [polar set](https://bliptext.com/articles/polar-set) of a bounded set is an [absolutely convex](https://bliptext.com/articles/absolutely-convex-set) and [absorbing set](https://bliptext.com/articles/absorbing-set).\nThe concept was first introduced by [John von Neumann](https://bliptext.com/articles/john-von-neumann) and [Andrey Kolmogorov](https://bliptext.com/articles/andrey-kolmogorov) in [1935](https://bliptext.com/articles/1935).\n\n# Definition\nSuppose X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over a [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K}.\nA subset B of X is called **' or just **' in X if any of the following equivalent conditions are satisfied:\n \n : For every [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) V of the origin there exists a real r > 0 such that for all scalars s satisfying |s| \\geq r. B is [absorbed](https://bliptext.com/articles/absorbing-set) by every [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) of the origin. \n For every neighborhood V of the origin there exists a scalar s such that \n For every neighborhood V of the origin there exists a real r > 0 such that for all scalars s satisfying |s| \\leq r. \n For every neighborhood V of the origin there exists a real r > 0 such that for all real \n Any one of statements (1) through (5) above but with the word \"neighborhood\" replaced by any of the following: \"[balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"open [balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"closed [balanced](https://bliptext.com/articles/balanced-set) neighborhood,\" \"open neighborhood,\" \"closed neighborhood\".\n For every sequence of scalars that converges to 0 and every sequence in B, the sequence converges to 0 in X. For every sequence in B, the sequence converges to 0 in X. \n Every [countable](https://bliptext.com/articles/countable-set) subset of B is bounded (according to any defining condition other than this one). \n \nIf \\mathcal{B} is a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) for X at the origin then this list may be extended to include:\n Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to \nIf X is a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) space whose topology is defined by a family \\mathcal{P} of continuous [seminorms](https://bliptext.com/articles/seminorm), then this list may be extended to include:\n p(B) is bounded for all \n There exists a sequence of non-zero scalars such that for every sequence in B, the sequence is bounded in X (according to any defining condition other than this one). \n For all B is bounded (according to any defining condition other than this one) in the [semi normed space](https://bliptext.com/articles/semi-normed-space) (X, p). \n B is weakly bounded, i.e. every continuous linear functional is bounded on B \n \nIf X is a [[norm](https://bliptext.com/articles/norm-mathematics)ed space](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-space) with [norm](https://bliptext.com/articles/norm-mathematics) \\|\\cdot\\| (or more generally, if it is a semi[[norm](https://bliptext.com/articles/norm-mathematics)ed space](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-space) and \\|\\cdot\\| is merely a semi[norm](https://bliptext.com/articles/norm-mathematics)), then this list may be extended to include:\n B is a norm bounded subset of By definition, this means that there exists a real number r > 0 such that for all b \\in B. \n \n B is a subset of some (open or closed) ball.\n \nIf B is a vector subspace of the TVS X then this list may be extended to include:\n B is contained in the closure of \\{0\\}.\n \nA subset that is not bounded is called.\n\n## Bornology and fundamental systems of bounded sets\nThe collection of all bounded sets on a topological vector space X is called the or the\nA or of X is a set \\mathcal{B} of bounded subsets of X such that every bounded subset of X is a subset of some\nThe set of all bounded subsets of X trivially forms a fundamental system of bounded sets of X.\n\n### Examples\nIn any [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) TVS, the set of closed and bounded [disks](https://bliptext.com/articles/absolutely-convex-set) are a base of bounded set.\n\n# Examples and sufficient conditions\nUnless indicated otherwise, a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) need not be [Hausdorff](https://bliptext.com/articles/hausdorff-space) nor [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space).\n \n Finite sets are bounded. \n Every [totally bounded](https://bliptext.com/articles/totally-bounded) subset of a TVS is bounded. \n Every [relatively compact set](https://bliptext.com/articles/relatively-compact-set) in a topological vector space is bounded.\nIf the space is equipped with the [weak topology](https://bliptext.com/articles/weak-topology-polar-topology) the converse is also true. \n The set of points of a [[Cauchy](https://bliptext.com/articles/cauchy-[net](https://bliptext.com/articles/net-mathematics)) sequence](https://bliptext.com/articles/cauchy-sequence) is bounded, the set of points of a [Cauchy](https://bliptext.com/articles/cauchy-[net](https://bliptext.com/articles/net-mathematics)) [net](https://bliptext.com/articles/net-mathematics) need not be bounded. \n The closure of the origin (referring to the closure of the set \\{0\\}) is always a bounded closed vector subspace.\nThis set is the unique largest (with respect to set inclusion ) bounded vector subspace of X. In particular, if is a bounded subset of X then so is \n \n**Unbounded sets**\nA set that is not bounded is said to be unbounded.\nAny vector subspace of a TVS that is not a contained in the closure of \\{0\\} is unbounded\nThere exists a [Fréchet space](https://bliptext.com/articles/fr-chet-space) X having a bounded subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any bounded subset of M.\n\n## Stability properties\n \n In any TVS, finite [unions](https://bliptext.com/articles/union-set-theory), finite [Minkowski sums](https://bliptext.com/articles/minkowski-sum), scalar multiples, translations, subsets, [closures](https://bliptext.com/articles/closure-topology), [interiors](https://bliptext.com/articles/interior-topology), and [balanced hulls](https://bliptext.com/articles/balanced-set) of bounded sets are again bounded. \n In any [locally convex TVS](https://bliptext.com/articles/locally-convex-topological-vector-space), the [convex hull](https://bliptext.com/articles/convex-set) (also called the [convex envelope](https://bliptext.com/articles/convex-envelope)) of a bounded set is again bounded.\nHowever, this may be false if the space is not locally convex, as the (non-locally convex) [Lp space](https://bliptext.com/articles/lp-space) L^p spaces for 0 \n The image of a bounded set under a [continuous linear map](https://bliptext.com/articles/continuous-linear-map) is a bounded subset of the codomain. \n A subset of an arbitrary [(Cartesian) product](https://bliptext.com/articles/product-space) of TVSs is bounded if and only if its image under every coordinate projections is bounded. \n If and X is a topological vector subspace of Y, then S is bounded in X if and only if S is bounded in Y. \n\n# Properties\nA [locally convex topological vector space](https://bliptext.com/articles/locally-convex-topological-vector-space) has a bounded neighborhood of zero [if and only if](https://bliptext.com/articles/if-and-only-if) its topology can be defined by a [seminorm](https://bliptext.com/articles/seminorm).\nThe [polar](https://bliptext.com/articles/polar-set) of a bounded set is an [absolutely convex](https://bliptext.com/articles/absolutely-convex-set) and [absorbing set](https://bliptext.com/articles/absorbing-set).\nUsing the definition of [[uniformly bounded](https://bliptext.com/articles/) sets](https://bliptext.com/articles/) given below, [Mackey's countability condition](https://bliptext.com/articles/) can be restated as: If are bounded subsets of a [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) then there exists a sequence of positive real numbers such that are [uniformly bounded](https://bliptext.com/articles/).\nIn words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.\n\n# Generalizations\n\n## Uniformly bounded sets\nA [family of sets](https://bliptext.com/articles/family-of-sets) \\mathcal{B} of subsets of a [topological vector space](https://bliptext.com/articles/topological-vector-space) Y is said to be in Y, if there exists some bounded subset D of Y such that\nwhich happens if and only if its union\nis a bounded subset of Y. In the case of a [normed](https://bliptext.com/articles/normed-space) (or semi[normed](https://bliptext.com/articles/normed-space)) space, a family \\mathcal{B} is uniformly bounded if and only if its union is norm bounded, meaning that there exists some real M \\geq 0 such that for every or equivalently, if and only if\nA set H of maps from X to Y is said to be if the family is uniformly bounded in Y, which by definition means that there exists some bounded subset D of Y such that or equivalently, if and only if is a bounded subset of Y. A set H of linear maps between two normed (or seminormed) spaces X and Y is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in X if and only if their [operator norms](https://bliptext.com/articles/operator-norm) are uniformly bounded; that is, if and only if\nAssume H is equicontinuous and let W be a neighborhood of the origin in Y. Since H is equicontinuous, there exists a neighborhood U of the origin in X such that for every h \\in H. Because C is bounded in X, there exists some real r > 0 such that if t \\geq r then\nSo for every h \\in H and every t \\geq r, which implies that Thus is bounded in Y. [Q.E.D.](https://bliptext.com/articles/q-e-d)\nLet W be a [balanced](https://bliptext.com/articles/balanced-set) neighborhood of the origin in Y and let V be a closed [balanced](https://bliptext.com/articles/balanced-set) neighborhood of the origin in Y such that\nDefine\nwhich is a closed subset of X (since V is closed while every h : X \\to Y is continuous) that satisfies for every h \\in H. Note that for every non-zero scalar n \\neq 0, the set n E is closed in X (since scalar multiplication by n \\neq 0 is a [homeomorphism](https://bliptext.com/articles/homeomorphism)) and so every C \\cap n E is closed in C.\nIt will now be shown that from which follows.\nIf c \\in C then H(c) being bounded guarantees the existence of some positive integer such that where the linearity of every h \\in H now implies thus and hence as desired.\nThus\nexpresses C as a countable union of closed (in C) sets.\nSince C is a [nonmeager subset](https://bliptext.com/articles/nonmeager-subset) of itself (as it is a [Baire space](https://bliptext.com/articles/baire-space) by the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem)), this is only possible if there is some integer n \\in \\N such that C \\cap n E has non-empty interior in C. Let be any point belonging to this open subset of C. Let U be any balanced open neighborhood of the origin in X such that\nThe sets form an increasing (meaning p \\leq q implies ) cover of the compact space C, so there exists some p > 1 such that (and thus ).\nIt will be shown that for every h \\in H, thus demonstrating that is uniformly bounded in Y and completing the proof.\nSo fix h \\in H and c \\in C. Let\nThe convexity of C guarantees z \\in C and moreover, z \\in k + U since\nThus which is a subset of Since n V is balanced and we have which combined with gives\nFinally, and imply\nas desired.\n[Q.E.D.](https://bliptext.com/articles/q-e-d)\nSince every [singleton subset](https://bliptext.com/articles/singleton-set) of X is also a bounded subset, it follows that if is an [equicontinuous](https://bliptext.com/articles/equicontinuity) set of [continuous linear operators](https://bliptext.com/articles/continuous-linear-operator) between two [topological vector spaces](https://bliptext.com/articles/topological-vector-space) X and Y (not necessarily [Hausdorff](https://bliptext.com/articles/hausdorff-space) or locally convex), then the [orbit](https://bliptext.com/articles/orbit-group-theory) of every x \\in X is a bounded subset of Y.\n\n## Bounded subsets of topological modules\nThe definition of bounded sets can be generalized to [topological modules](https://bliptext.com/articles/topological-module).\nA subset A of a topological module M over a [topological ring](https://bliptext.com/articles/topological-ring) R is bounded if for any neighborhood N of 0_M there exists a neighborhood w of 0_R such that","revisionCount":0},"hasMore":false,"nextCursor":null}

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