# Examples of Banach spaces and their duals

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 $X$?

• The $\operatorname{Sym}$ 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.) Apr 16, 2010 at 2:30
• Just to add to Harald's comment: of course this doesn't determine the dual space up to isometry, but up to (linear, bicontinuous) isomorphism. In many contexts one only cares about the Banach space up to isomorphism, but occasionally one might wish to determine the norm more precisely. Apr 16, 2010 at 2:36
• Note ... you mean Radon measures on [0,1], not on X . Apr 16, 2010 at 12:28

In many cases, you simply work through or unwind the definition of a dual Banach space, namely the vector space of bounded linear functionals. In the specific case you're interested in, you simply get a space of $Sym^*$-valued distributions (in the sense of Laurent Schwartz).
• Since $U$ is compact and we're dealing with a fixed degree of differentiability, I don't think we get distributions per se. My guess would be `$Sym*$'-valued complex measures on $U$. Apr 16, 2010 at 2:37
• Well, they're not smooth distributions. But they're not just measures, either, because if $x_0 \in U$, the dual space contains the functional $f \mapsto \langle A,\partial_i\partial_jf(x_0)\rangle$, where $A \in Sym^*$. Apr 16, 2010 at 11:07