Corollary to the Banach-Steinhaus Theorem

Question

Let $X$ be a normed vector space and let $Y\subseteq X$.

If $\sup_{y\in Y}\left|f\left(y\right)\right|<\infty$ for every $f\in X^*$, then $\sup_{y\in Y}\left\|y\right\|<\infty$.

Could someone please provide a hint? I feel like this should be super straightforward.

Answer 1

Given $y\in Y$, consider the function $\hat y\in X^{**}$ given by $$\hat y(f)=f(y).$$ Your hypothesis is that $$ \sup\{|\hat y(f)|:\ y\in Y\}<\infty. $$ By the Uniform Boundedness Principle, $$ \sup\{\|\hat y\|: \ y\in Y\}<\infty. $$ The last ingredient is the equality $\|\hat y\|=\|y\|$. This follows directly from the well-known equality $$ \|y\|=\sup\{|f(y)|:\ f\in X^*,\ \|f\|=1\}. $$