Application of Banach Steinhaus theorem

Question

Let $X,Y$ be Banach spaces and let $A:X\rightarrow Y,B:Y^*\rightarrow X^*$ be linear maps.

Show that if for all $x\in X,y\in Y^*,y^*(Ax)=By^*(x)$, then both $A$ and $B$ are continuous

Answer 1

Noting that for a fixed $x$ and $\|y^*\| \le 1$, the map $y^* \mapsto y^*(Ax)$ is bounded uniformly by $\|Ax\|$.

Then Banach Steinhaus gives $\sup_{\|x\| \le 1, \|y^* \| \le 1} |y^*(Ax)| < \infty$.

We have $\|B\| = \sup_{\|y^*\| \le 1} \|B y^*\| = \sup_{\|x\| \le 1, \|y^*\| \le 1} |B y^*(x)| = \sup_{\|x\| \le 1, \|y^*\| \le 1} |y^*(Ax)| $, hence we see that $B$ is bounded.

The other direction follows directly from this bound.