$X$ Banach space, and $A: X \rightarrow X'$ is a lineal operator. If $(Ax)(y)=(Ay)(x)$ prove that $A$ is a continuous operator.

Question

I am trying to prove that if $X$ is a Banach space, $A: X \rightarrow X'$ $(X'$ is the dual) is a linear operator, and $(Ax)(y)=(Ay)(x)$ then $A$ is a continuous operator.

I have no idea how can i find that $M$ to prove the continuity with this conditions. Any ideas? Thanks

Answer 1

Suppose $x_n\to x$ in $X$ and $A(x_n)\to f$ in $X'$. Then for all $y\in X$ you have that: $$f(y)=\lim_n (Ax_n)\ (y) = \lim_n(Ay)\ (x_n) = (Ay)\ (x) = (Ax)\ (y)$$ and hence $f= A(x)$. This implies that $A$ is a closed operator. Since $A$ has domain and codomain a Banach space by the closed graph theorem $A$ is continuous.