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Let $E$ be a normed space over field $\mathbb{C}$ and $A$ a subset of $E$ such that $\varphi[A]$ is bounded for all $\varphi\in E'$. How to prove that $A$ is bounded applying Hahn-Banach Theorem and Uniform Boundedness Principle?

Any hint would be appreciated.

felipeuni
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    Hint: identify $E$ with a subset of $E''$. Then note that $A$ is a pointwise bounded set of continuous functionals on $E'$ and use the uniform boundedness principle. – Nate Eldredge Jul 10 '19 at 01:57

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Consider the standard embedding $\varphi\colon E\mapsto E^{\ast\ast}$, where $\varphi(e)=\mathrm{ev}_e\colon E^{\ast}\to\Bbb R$, given by $\mathrm{ev}_e(\psi)=\psi(e)$ for all $\psi\in E^\ast$.

Now you can identify $A$ with $\varphi(A)\subseteq E^{\ast\ast}$ and note that $\mathrm{ev}_a\colon E^\ast\to\Bbb R$ is bounded for every $a\in A$, by hypothesis. By the uniform boundedness principle there is some $c\in\Bbb R$ with $\|\mathrm{ev}_a(\psi)\|\leq c\|\psi\|$ for all $a\in A$ and all $\psi\in E^\ast$.

But remembering that $$\|a\|=\sup_{\substack{\psi\in E^\ast \\ \|\psi\|=1}}\psi(a)$$ we get $\|a\|\leq c$ for all $a\in A$, so $A$ is bounded.

Alessandro Codenotti
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Hint:By Hahn Banach for every $x$, you can extend $f_x(x)=\|x\|$ defined on $Vect(x)$ to $E$

Tsemo Aristide
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