Examples of Banach spaces and their duals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:49:12Z http://mathoverflow.net/feeds/question/21524 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21524/examples-of-banach-spaces-and-their-duals Examples of Banach spaces and their duals Tom LaGatta 2010-04-16T01:29:38Z 2011-08-08T21:24:08Z <p>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?</p> <p>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 $X$?</p> http://mathoverflow.net/questions/21524/examples-of-banach-spaces-and-their-duals/21526#21526 Answer by Deane Yang for Examples of Banach spaces and their duals Deane Yang 2010-04-16T01:51:08Z 2010-04-16T01:51:08Z <p>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).</p> http://mathoverflow.net/questions/21524/examples-of-banach-spaces-and-their-duals/21558#21558 Answer by Gerald Edgar for Examples of Banach spaces and their duals Gerald Edgar 2010-04-16T12:27:41Z 2010-04-16T12:27:41Z <p>Find many worked-out examples, all the important ones, in Dunford &amp; Schwartz, volume I. See the top row of the table beginning on page 374.</p> http://mathoverflow.net/questions/21524/examples-of-banach-spaces-and-their-duals/72403#72403 Answer by dan232 for Examples of Banach spaces and their duals dan232 2011-08-08T21:24:08Z 2011-08-08T21:24:08Z <p>In wikipedia, there is a <a href="http://en.wikipedia.org/wiki/List_of_Banach_spaces" rel="nofollow">list of banach spaces</a> with its dual space and it also tells you if it is reflexive, for example.</p>