I can't make up my mind whether this question is trivial, or simply wrong, so i decided to ask, just in case someone sees a fallacy in my reasoning:
Question: Suppose $V,W$ are two banach spaces, and $T:V\to W$ is an isomorphism. Is $T^*:W^*\to V^*$ an isomorphism?
On the one hand this seems trivial- it requires a little work, but one can show that $T^*$ is injective, given that $T$ is surjective without working too hard, so if $T^*$ is also surjective, the open mapping theorem should finish the work for us:
To show this, let $f\in V^*$ be arbitrary. Then $f\circ T^{-1}:W\to \mathbb{C}$ is bounded and linear (since $T^{-1}$ and $f$ both are), and $T^*(f\circ T^{-1})(w)=f\circ T^{-1}(Tw)=f(w)$
again, this apears (to me, at least) to be correct, but my little experience with Banach spaces has taught me to fear such immediate results, when discussing duals :-P...
anyhow, I would be very happy if someone could tell me if I'm correct, or otherwise, give a counter-example, or point to a mistake.
Additionally, assuming this isn't as immediate as I thought- does the assertion hold when $T$ is an isometric isomorphism?
Thank you very much :-)
(p.s i added the homework tag, as this question arose as part of a h.w assignment, but this isn't a h.w question per-se)