Are linear and continuous mappings between locally convex vector spaces bounded?

Question

I know that continuity and boundedness of linear mappings between normed vector spaces are equivalent, but does the same hold true for locally convex vector spaces? If so, how can we prove it?

Answer 1

I believe the answer is yes - let's see if I can prove it. First we need to define "bounded".

Recall that $S\subset X$ is said to be bounded if for every open set $V$ containing the origin there exists $c>0$ so that $S\subset cV$.

Now given two topological vector spaces $X$ and $Y$ we say that $T:X\to Y$ is bounded if $T(S)$ is bounded in $Y$ for every bounded $S\subset S$.

Say $T:X\to Y$ is linear and continuous. Say $S\subset X$ is bounded; we need too show that $T(S)$ is bounded in $Y$. So say $V$ is a neighborhood of the origin in $Y$. Let $W=T^{-1}(V)$. Since $S$ is bounded there exists $c>0$ with $S\subset cW$. So $T(S)\subset T(cW)=cT(W)\subset cV$. Sure enough, $T$ is bounded.

Offhand I don't see how to prove the converse in general. But in locally convex spaces it must be easy. Say the topology on $X$ is defined by a family of seminorms $A$. Then $S\subset X$ is bounded if and only if every $\rho\in A$ is bounded on $S$. It follows that if the topology on $Y$ is defined by a family of seminorms $B$, then the linear map $T :X\to Y$ is bounded if and only if for every $\rho\in B$ there exist $\rho_1,\dots,\rho_n\in A$ with $$\rho(Tx)\le c\sum_{j=1}^n\rho_j(x).$$ It's easy to see that that condition implies that $T$ is continuous.