If we identify $V$ and $U$ with their canonical images in $V^{**}$ and $U^{**}$ prove that the restriction of $T^{**}$ to $V$ coincides with $T$.

Question

Let $T : V \rightarrow U$ be a bounded map between two normed spaces. Let $T^* : U^* \rightarrow V^*$ be defined by $T^*(f) = f\circ T$ for all $f\in U^*$ (the adjoint map).

My Question: If we identify $V$ and $U$ with their canonical images in $V^{**}$ and $U^{**}$ prove that the restriction of $T^{**}$ to $V$ coincides with $T$.

I think I want to try and use the canonical mappings $V \rightarrow V^{**}$ and $U \rightarrow U^{**}$ but I am not entirely sure what the canonical images would be. The restriction would be $T^{**}\cap V$ and I am not sure where that comes into play...

I would really appreciate some help on this proof. hank you.

Answer 1

I will instead write $v^{**}\in V^{**}$ and $u^{**}\in U^{**}$ as the images of $v\in V$ and $u\in U$, respectively, under the duality maps $V\to V^{**}$ and $U\to U^{**}$. We note that $$T^{**}v^{**}(\varphi)=v^{**}\circ T^*(\varphi) =v^{**}(\varphi \circ T)=\varphi\circ T(v)=\varphi(Tv)=(Tv)^{**}(\varphi)$$ for all $v\in V$ and $\varphi \in U^*$. Therefore, $T^{**}v^{**}=(Tv)^{**}$ for every $v\in V$, and the conclusion follows.