Examples of Banach spaces and their duals - MathOverflow

Examples of Banach spaces and their duals

2010-04-16T01:29:38Z

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 $X$?

Answer by Deane Yang for Examples of Banach spaces and their duals

2010-04-16T01:51:08Z

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).

Answer by Gerald Edgar for Examples of Banach spaces and their duals

2010-04-16T12:27:41Z

Find many worked-out examples, all the important ones, in Dunford & Schwartz, volume I. See the top row of the table beginning on page 374.

Answer by dan232 for Examples of Banach spaces and their duals

2011-08-08T21:24:08Z

In wikipedia, there is a list of banach spaces with its dual space and it also tells you if it is reflexive, for example.