Banach spaces and isomorphism between dual spaces

Question

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?

Answer 1

Every normed vector space $X$ extends to its completion $\tilde{X}$, which is then a Banach space (it is unique up to isomorphisms). If $X$ and $Y$ are isomorphic then $\tilde{X}$ and $\tilde{Y}$ are also isomorphic. Now the dual space $X^*$ is the same as $\tilde{X}^*$ because the functionals on $X$ extend by continuity to $\tilde{X}$. So it follows that if $X$ and $Y$ are isomorphic, so are $\tilde{X}$ and $\tilde{Y}$, and hence also their duals $\tilde{X}^*=X^*$ and $\tilde{Y}^*=Y^*$.

Answer 2

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 $T^{*}$.