Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces and $(L_b(X,Y),||.||)$ be the space of all bounded linear functions from $X \rightarrow Y$. Let $x \in X$ fixed. How can I show that the function
$$(L_b(X,Y)) \rightarrow Y, A \mapsto Ax$$ is bounded? I could already prove it is linear so if that is needed to show that it is bounded, you can assume it
Recall the definition of the norm on $L_b(X,Y)$: $$\|A\|=\sup\{\|Ax\|:x\in X, \|x\|\leq1\}$$ (or something equivalent to this). Using this definition, you can show that if $x\in X$, we have $$\|Ax\|\leq\|A\|\|x\|.$$ Thus, if we let $T_x:L_b(X,Y)\to Y$ denote the map $A\mapsto Ax$, there is some $M$ (namely $\|x\|$) such that for all $A\in L_b(X,Y)$ we have $\|T_xA\|=\|Ax\|\leq M\|A\|$.