Let $X$ be banach space, $Y$ be normed space, $\mathcal A \subset \mathcal B(X, Y)$ be some set of continuous linear operators $X \to Y$. I need to prove that if
$$\forall x \in X, g \in Y^* \;\; \sup_{A \in \mathcal A} |g(Ax)| < \infty,$$
then
$$\sup_{A \in \mathcal A} ||A|| < \infty.$$
I have managed to use uniform boundedness principle to deduce
$$\forall g \in Y^* \sup_{A \in \mathcal A} ||g \circ A|| <\infty.$$
I have no idea how to proceed.
Best Answer
Define $T_A: Y^{*} \to X^{*}$ by $T_A (g)=g\circ A$. For each $g$, $(T_Ag)_{A\ \in \mathcal A}$ is norm bounded (from what you have already observed). By Uniform Boundeness Principle we get $\sup_{A \in \mathcal A, \|g\|\leq 1} \|g\circ A\| <\infty$. This means $\sup \{\|A\|:A \in \mathcal A\}<\infty$.