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.
When linear map is bounded and bijective then $X$ is reflexive.
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functional-analysis
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1What are $X$ and $Y$? – 2012-12-12
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0Taking the second dual gives $T^{ **}:X^{ **}\rightarrow Y^{ **}=Y$. – 2012-12-12
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0@paulgarrett can you tell me what is $T^{**}$ ? – 2012-12-12
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0@DavideGiraudo : I edited . – 2012-12-12
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0Ok, so what do you know about reflexivity? – 2012-12-12
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0@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 $T^{**}:X^{**}\rightarrow Y^{**}$...