Tim wrote:

> The dual space will still be a Banach space, but a Banach space does not need to be isomorphic to its dual (those who are are called reflexive).

No, a Banach space $V$ is [reflexive](http://en.wikipedia.org/wiki/Reflexive_space) if the canonical map $i : V \to V^{**}$ is an isomorphism. So for example $L^p(X)$ is reflexive for $1 \lt p \lt \infty$ when $X$ is a $\sigma$-finite measure space, but it's only isomorphic to its dual when $p = 2$.