I'm studying fuctional analysis and specifically reflexive spaces. My textbook has a introductory level, so don't cover so many things. My questions are:
1) If $X$ and $Y$ are isomorphics and $X$ is reflexive, is $Y$ reflexive?
2) Trying to solve the previous question, I was wondering if $X$ and $Y$ are isomorphics, are $X^{*}$ and $Y^{*}$ isomorphics? ( I know it's true for finite dimensional spaces). Should I use adjoint operator to solve it?
Thanks.
The answer to both questions is yes.
Let's adress the second question first. Let $A : X \rightarrow Y$ is a bijective bounded linear operator with bounded inverse $B=A^{-1}: Y \rightarrow X$. We have: $$ \text{id}_{X} = BA, \quad \text{id}_{Y} = AB $$ and therefore $$ \text{id}_{X^\ast} = A^\ast B^\ast, \quad \text{id}_{Y^\ast} = B^\ast A^\ast $$ Hence $A^\ast$ is invertible and has as bounded inverse $B^\ast$.
For Q1: If $X$ and $Y$ are isomorphic via $A: X \rightarrow Y$, the above argument applied to $A^{\ast}$, shows that the bidual operator $A^{\ast \ast} X^{\ast\ast} \rightarrow Y^{\ast \ast}$ is an isomorphism. Let $\iota_X: X \rightarrow X^{\ast \ast}$ and $\iota_Y: Y \rightarrow Y^{\ast \ast}$ denote the cannonical inclusions. One checks that $\iota_Y \circ A = A^{\ast \ast} \circ \iota_X$.
This implies that if $\iota_Y$ is an isomorphism, then so is $\iota_X$ and vice versa.