Dualization map is surjective I am practicing for my exam and I want to solve the following problem. 
Let $X,Y$ be normed reflexive spaces. Show that the "Dualization map" $':B(X,Y)\to B(Y',X')$,  $T\mapsto T'$ is surjective
I want to use reflexivity since $X''$ and $X$, likewise for $Y$, are isometrically isomorphic. But I get stuck. Has anybody an idea?

Hint: for every $U'\in B(Y',
X') $
Consider $U"=(U')':X"=X\rightarrow Y"=Y$. 
