Lat $X,Y$ be banach spaces and $T:X\to Y$ a function. Assume that there is a function $S:Y^*\to X^*$ such that for all $x\in X, y^*\in Y^*$ , $y^*(Tx)=(Sy^*)(x)$. Prove that $T\in B(X,Y)$ (that means $T:X\to Y$ is linear and bounded).
I have a problem with this question since i need to show that T is linear and bounded however I did nof understand how to use the information given about S! Can you please clarify it to me.
Many thanks.
We have that $$y^*(T(x + y)) = (Sy^*)(x + y) = (Sy^*)(x) +(Sy^*)(y) = y^*(Tx) + y^*(Ty) = y^*(Tx + Ty).$$ Since $y^* \in Y^*$ is arbitrary, it follows that the arguments are equal (by Hahn-Banach theorem). Something similar holds for scalar multiplication.
If $x_n \to 0$ and $Tx_n \to y$, then $y^*(Tx_n) \to y^*(y)$ and $(Sy^*)(x_n) \to (Sy^*)(0) = 0$, since $Sy^* \in X^*$. It follows that $y = 0$, hence $T$ is bounded by the closed graph theorem.