# Application of Banach-Steinhaus?

Following problem in an exam last week:

Let $X$ be a complex normed space and $S\subset X$ an arbitrary subset. Show that if $\{ x^*(x):x\in S \}$ is a bounded set in $\mathbb{C}$ for each $x^* \in X^*$ then there exists a $K$ s.t. $\vert\vert x\vert\vert\leq K, \forall x\in S$.

I was not able to solve it. Is it an application of Banach-Steinhaus?

The idea is to use the canonical injection $\delta :X\hookrightarrow X^{**},x\mapsto (x^*\mapsto x^*(x))$ which is isometric. Then the family $\{\delta(x):x\in S\}$ is pointwise bounded thus by Banach-Steinhaus bounded in the operator norm on the bidual space and since $\delta$ is isometric it is bounded in $X$ itself.
• Sorry. You are right . You definitely need this space to be complete to apply Banach-Steinhaus. You might consider a completion of $X$. – Peter Melech Mar 25 '17 at 12:01
• I think it is, You prove the statement for a completion and conclude that the set is already bounded in $X$ – Peter Melech Mar 25 '17 at 12:04
• but what is the completion of $X$? – tubmaster Mar 25 '17 at 12:05