Suppose V is a normed vector space and B is a subset of V such that phi(B) is bounded for every phi in the dual space of V, prove that B is bounded

Question

More formally:

Suppose V is a normed vector space and B is a subset of V such that $$sup_{x\in B}|\phi(x)|<\infty$$ $\forall \phi \in V'$ where $V'$ is the dual space of $V$ (the normed vector space consisting of the bounded linear functionals on $V$).

Prove that $$sup_{x\in B}||x||<\infty$$

I am thinking of doing this by proving that if $sup_{x\in B}||x||=\infty$, then this implies $\exists \phi'\in V'$ s.t. $$sup_{x\in B}|\phi'(x)|=\infty$$ Is this a good way to proceed? I am a little lost so any hints would be appreciated!

Answer 1

It is not that simple. This requires Uniform Boundednss Principle (alias Banach-Steinhaus Theorem). For $x \in B$ define $F_x: V'\to F$ by $F_x(\phi)=\phi (x)$. Then $\|F_x\|=\|x\|$ and $F_x(\phi),x \in B$ is bounded for each fixed $\phi$. So Uniform Boundednss Principle finishes the proof.