There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued continuous functions on $[0,1]$, then $X^*$ is the space of Radon measures on $[0,1]$. When you are confronted with some Banach space, where do you go to figure out a representation of its dual space? Is there a book or survey article with a rich set of examples?

Here is the particular example which motivates this question. Let $\operatorname{Sym}$ be the space of symmetric $n \times n$ real matrices with a usual matrix norm. Let $U \subseteq \mathbb R^n$ be compact, and let $X = C^{2+\alpha}(U, \operatorname{Sym})$ with the obvious norm. What is the dual space of $X$?

`$\operatorname{Sym}$`

is a red herring here. Since`$X=C^{2+\alpha}(U)\otimes\operatorname{Sym}$`

, you get`$X^*=C^{2+\alpha}(U)^*\otimes\operatorname{Sym}^*$`

. The same goes if`$\operatorname{Sym}$`

is replaced by any finite-dimensional space. (The tensor product of two infinite dimensional Banach space is an entirely different kettle of fish, of course.) – Harald Hanche-Olsen Apr 16 '10 at 2:30