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I am looking for the proof of : If $T$ is bounded ,bijective linear map from $X$ to $Y$ where $X$ and $Y$ are Banach spaces, then $X$ is reflexive if and only if $Y$ is reflexive . Any suggestions are appreciated.

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    What are $X$ and $Y$?2012-12-12
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    Taking the second dual gives $T^{ **}:X^{ **}\rightarrow Y^{ **}=Y$.2012-12-12
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    @paulgarrett can you tell me what is $T^{**}$ ?2012-12-12
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    @DavideGiraudo : I edited .2012-12-12
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    Ok, so what do you know about reflexivity?2012-12-12
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    @David Giraudo : the fact that canonical embedding is isomorph from Banach space to its dual. But how do i go about dealing with the linear transformation. I haven't yet learnt many theorems about reflexivity . But i guessed that it doesn't require more results .2012-12-12

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To amplify slightly my comment: given $T:X\rightarrow Y$, the adjoint $T^*:Y^*\rightarrow X^*$ is characterized by $(T^*\mu)(x)=\mu(Tx)$ for $x\in X$ and $\mu\in Y^*$. Doing this one more time gives $T^{**}:X^{**}\rightarrow Y^{**}$...