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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?

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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.

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  • $\begingroup$ nice, but I got a litte problem: is banach steinhaus possible, since we do not have to do with a Banach space here $\endgroup$ – tubmaster Mar 25 '17 at 11:52
  • $\begingroup$ Sorry. You are right . You definitely need this space to be complete to apply Banach-Steinhaus. You might consider a completion of $X$. $\endgroup$ – Peter Melech Mar 25 '17 at 12:01
  • $\begingroup$ what do you mean by "consider a completion"? is the assertion not true in our setting? $\endgroup$ – tubmaster Mar 25 '17 at 12:03
  • $\begingroup$ I think it is, You prove the statement for a completion and conclude that the set is already bounded in $X$ $\endgroup$ – Peter Melech Mar 25 '17 at 12:04
  • $\begingroup$ but what is the completion of $X$? $\endgroup$ – tubmaster Mar 25 '17 at 12:05
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You can see Brezis's book...corollary 2.4 enter image description here

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