Equicontinuous family of continuous linear maps

Definition

Let $E$ and $F$ be topological vector spaces, and let $\mathcal T$ be a family of continuous linear maps $T:E\to F$ . The family $\mathcal T$ is equicontinuous if, for every neighborhood $V$ of $0$ in $F$ , there is a neighborhood $U$ of $0$ in $E$ such that

T(U)\subseteq V \qquad\text{for every }T\in\mathcal T.

The same source neighborhood must therefore control all operators in the family. For linear maps, this condition at $0$ is equivalent to equicontinuity at every point.

Seminorm criterion

Suppose $E$ and $F$ are locally convex. Equicontinuity is equivalent to the following: for every continuous seminorm $q$ on $F$ , there are continuous seminorms $p_1,\ldots,p_n$ on $E$ and $C>0$ such that

q(Tx)\le C\max_{1\le j\le n}p_j(x)

for all $T\in\mathcal T$ and $x\in E$ . This is the locally convex analogue of a uniform operator-norm bound.

Duality and uniform boundedness

For a family $\mathcal T$ in the topological dual $E'$ , equicontinuity means that some zero-neighborhood $U\subseteq E$ satisfies $\lvert\varphi(x)\rvert\le1$ for every $\varphi\in\mathcal T$ and $x\in U$ ; equivalently, $\mathcal T$ lies in the polar of $U$ . If $E$ is barreled, pointwise bounded families of continuous linear maps from $E$ into a locally convex space are equicontinuous Trèves, Chapter 32.

Stability and cautions

Subfamilies, finite unions, and the balanced convex hull of an equicontinuous family remain equicontinuous. Composing on either side with a fixed continuous linear map also preserves equicontinuity. Equicontinuity is stronger than requiring each member to be continuous. It is also distinct from equicontinuity of arbitrary maps between metric spaces, although the two definitions agree for linear maps when translated into zero-neighborhood language.

References

François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967; Dover reprint, 2006. Dover publisher record. Relevant: Chapter 32 on equicontinuity and uniform boundedness.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter III on spaces of continuous linear mappings.