Show that $L\in\mathcal{L}(V,W)$.

Question

Let $V$ and $W$ be two Banach spaces, $L:V\rightarrow W$ be a linear map.

Suppose for any $f\in W'$, the dual of $W$, $f(Lv)$ is a bounded linear functional on $V$.

Show that $L\in\mathcal{L}(V,W)$.

Answer 1

Let $B$ be the closed unit ball in $V$. For $x\in B$ define $F_x : W'\to\mathbb R$ by $F_x(f) := f(Lx)$. We have $|F_x(f)|\le\|f\|\|Lx\|$, hence $F_x\in W''$ with $\|F_x\|\le\|Lx\|$. By Hahn-Banach, there exists $f_x\in W'$ such that $\|f_x\|=1$ and $f_x(Lx) = \|Lx\|$. We conclude $|F_x(f_x)| = |f_x(Lx)| = \|Lx\|$. This shows $\|F_x\| = \|Lx\|$.

Now, for fixed $f\in W'$ we have that $\sup_{x\in B}|F_x(f)| = \sup_{x\in B}|f(Lx)| < \infty$ by assumption. The principle of uniform boundedness now asserts that $\sup_{x\in B}\|F_x\| < \infty$ and thus $\sup_{x\in B}\|Lx\| < \infty$. But this means exactly that $L$ is bounded.