Bounded subset of a topological vector space - Imported Knowlpedia Content

Definition

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$ . A subset $B\subseteq X$ is bounded if every neighborhood $U$ of $0$ absorbs $B$ : there is $r>0$ such that

B\subseteq tU

for every scalar $t$ with $\lvert t\rvert\geq r$ . Equivalently, for every net of nonzero scalars $t_i$ with $\lvert t_i\rvert\to\infty$ , the rescaled sets $t_i^{-1}B$ eventually lie in each zero-neighborhood. This is also called von Neumann boundedness. It depends on the vector-space topology and need not arise from any metric or norm.

Comparison with metric boundedness

In a normed vector space, this definition is equivalent to $\sup_{x\in B}\lVert x\rVert<\infty$ : apply the definition to the open unit ball in one direction, and use balls as a zero-neighborhood basis in the other. A general topological vector space may have no distinguished norm, so a statement about finite diameter would not be intrinsic.

Seminorm criterion

If $X$ is locally convex, then $B$ is bounded exactly when

\sup_{x\in B}p(x)<\infty

for every continuous seminorm $p$ on $X$ . This criterion turns an apparently topological condition into a family of scalar estimates. It is particularly useful for function spaces whose topologies are specified by many seminorms.

Stability properties

Finite subsets, convergent sequences together with their limits, and compact subsets are bounded. Finite unions, sums, scalar multiples, and subsets of bounded sets remain bounded. Every continuous linear map sends bounded sets to bounded sets. The converse implication does not characterize continuity without extra hypotheses on the source space.

The neighborhood and seminorm formulations, together with these permanence properties, are developed systematically in Schaefer and Wolff, Chapters I–II.

References

Nicolas Bourbaki, Topological Vector Spaces: Chapters 1–5, Springer, 2003. Springer DOI record. Relevant: Chapters I–III.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapters I–II.