A problem on dual spaces

Question

Let $ (X,\Vert \cdot\Vert_X)$ be a Banach space and let $Y\subset X$. Assume for any $ g\in X' $ the set$$ \{g(y):y\in Y\} $$ is bounded. Show that the set Y is bounded

Please help! I spent an hours on this problem but has no clue!

Answer 1

Use the Banach-Steinhaus theorem in $X''$. By viewing $y \in Y$ as a linear operator on $X'$ by the rule $y: X' \to \mathbb C$, $$u \in X' \mapsto u(y) \in \mathbb C.$$ The Banach-Steinhaus theorem shows that the set of operator norms $$\{\| y \|_{X' \to \mathbb C}\}_{y\in Y}$$ is bounded, and then $\|y\|_X = \|y \|_{X' \to \mathbb C}$ shows that $Y$ is bounded.