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

This question is an exact duplicate of:

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.

## marked as duplicate by Arnaud D., Lee David Chung Lin, user10354138, ArsenBerk, Don ThousandNov 8 '18 at 17:07

This question was marked as an exact duplicate of an existing question.

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.