Continuous linear map between topological vector spaces

Definition

Let $E$ and $F$ be topological vector spaces over the same field. A continuous linear map $T:E\to F$ is a linear map that is continuous for the given topologies. Linearity makes continuity at one point equivalent to continuity everywhere. In particular, $T$ is continuous exactly when, for every neighborhood $V$ of $0$ in $F$ , there is a neighborhood $U$ of $0$ in $E$ such that

T(U)\subseteq V.

Thus continuity compares the chosen topologies on the source and target; it is not an algebraic property of the underlying linear map.

Normed-space specialization

If $E$ and $F$ are normed vector spaces, continuity is equivalent to the existence of $C\geq 0$ such that

\lVert Tx\rVert_F\leq C\lVert x\rVert_E \qquad (x\in E).

This familiar bounded-operator criterion is a special feature of norm topologies. For general topological vector spaces, continuity is expressed with zero-neighborhoods or families of seminorms rather than one operator norm.

Locally convex criterion

Suppose $E$ and $F$ are locally convex. For every continuous seminorm $q$ on $F$ , continuity of $T$ implies that there are continuous seminorms $p_1,\ldots,p_n$ on $E$ and $C>0$ with

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

Conversely, such estimates for a defining family of seminorms on $F$ imply continuity. This is the standard form used for spaces of smooth functions and distributions.

Categorical role

Identity maps and composites of continuous linear maps are continuous and linear. Topological vector spaces with these maps therefore form a category. A linear bijection need not be an isomorphism in this category: its inverse must also be continuous.

The zero-neighborhood criterion and the theory of spaces of continuous linear mappings are treated in Bourbaki, Chapters I and III.

References

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