Topology generated by a family of seminorms - Imported Knowlpedia Content

Definition

Let $E$ be a vector space and let $\mathcal P$ be a family of seminorms on $E$ . The topology generated by $\mathcal P$ has a neighborhood basis at $0$ consisting of the sets

U(p_1,\ldots,p_n;\varepsilon) =\{x\in E:p_j(x)<\varepsilon\text{ for }1\leq j\leq n\},

where $p_1,\ldots,p_n\in\mathcal P$ , $n<\infty$ , and $\varepsilon>0$ . Neighborhoods at $x$ are translates $x+U$ . This is the coarsest vector topology making every $p\in\mathcal P$ continuous, and it makes $E$ a locally convex space. It is Hausdorff exactly when the seminorms separate points.

Convergence

A net $x_\alpha$ converges to $x$ in the generated topology exactly when

p(x_\alpha-x)\longrightarrow0 \qquad\text{for every }p\in\mathcal P.

Sequences suffice only when the topology is first countable, so nets cannot generally be replaced by sequences for an uncountable seminorm family.

Hausdorffness and metrizability

The topology is Hausdorff precisely when

\bigcap_{p\in\mathcal P}\ker p=\{0\}.

If a countable family $(p_n)$ generates the topology, then it is metrizable by, for example,

d(x,y)=\sum_{n=1}^{\infty}2^{-n} \frac{p_n(x-y)}{1+p_n(x-y)}.

Metrizability does not imply completeness; that is a separate condition on Cauchy nets or sequences.

Comparison and examples

If $\mathcal P\subseteq\mathcal Q$ , then the topology generated by $\mathcal Q$ is finer than the one generated by $\mathcal P$ . A single norm generates the usual norm topology. Evaluation seminorms generate pointwise-convergence topologies, while suprema of derivatives on compact sets generate the standard topologies on spaces of smooth functions.

References

Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: §1.37.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier book record. Relevant: Chapter 7.