Maybe is a silly question, but I have got a doubt about it:
Let $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ be two linearly isomorphic Banach spaces. Then we know that their dual spaces are linearly isomorphic as well.
Is the hypothesis of being Banach necessary?
Can't I simply take normed vector spaces?
Best Answer
If by 'linearly isomorhic' you mean existence of a linear bijection $T$ such that $T$ and $T^{-1}$ are both bounded, the the answer is yes, completeness is not necessary. This is because if $T$ is such an isomorphism then so is the adjoint $T^{*}$.