Bounded set (topological vector space)

For bounded sets in general, see bounded set.

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. Conversely a set that is not bounded is called unbounded.

Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Definition

Given a topological vector space (X,τ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α such that

S\subseteq \alpha N

with

\alpha N:=\{\alpha x\mid x\in N\}

This is equivalent^[1] to the condition that S is absorbed by every neighborhood of the zero vector, i.e., that for all neighborhoods N, there exists t such that

|\alpha |\geq t\Rightarrow S\subseteq \alpha N

Bounded subsets of a topological vector space over the real or complex field can also be characterized by their sequences, for S is bounded in X if and only if for all sequences (c_n) of scalars converging to 0 and all (similarly-indexed) countable subsets (x_n) of S, the sequence of their products (c_n x_n) necessarily converges to zero in X.

In locally convex topological vector spaces the topology τ of the space can be specified by a family P of semi-norms. An equivalent characterization of bounded sets in this case is, a set S in (X,P) is bounded if and only if it is bounded for all semi normed spaces (X,p) with p a semi norm of P.

Examples and nonexamples

Every finite set of points is bounded
The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not to be bounded.
Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
A (non null) subspace of a Hausdorff topological vector space is not bounded

Properties

The closure of a bounded set is bounded.
In a locally convex space, the convex envelope of a bounded set is bounded. (Without local convexity this is false, as the $L^{p}$ spaces for $0<p<1$ have no nontrivial open convex subsets.)
The finite union or finite sum of bounded sets is bounded.
Continuous linear mappings between topological vector spaces preserve boundedness.
A locally convex space is seminormable if and only if there exists a bounded neighbourhood of zero.
The polar of a bounded set is an absolutely convex and absorbing set.
A set A is bounded if and only if every countable subset of A is bounded

Generalization

The definition of bounded sets can be generalized to topological modules. A subset A of a topological module M over a topological ring R is bounded if for any neighborhood N of 0_M there exists a neighborhood w of 0_R such that w A ⊂ N.

Notes

↑ Schaefer 1970, p. 25.

References

Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 44–46.
H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 25–26. ISBN 0-387-05380-8.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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