Double duals characteristic - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T10:18:40Z http://mathoverflow.net/feeds/question/106695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106695/double-duals-characteristic Double duals characteristic student 2012-09-08T23:41:23Z 2012-09-09T23:23:24Z <p>Recall that (for $1\le p&lt;\infty$), <code>$\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$</code>, with norm <code>$||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$</code>.</p> <p>It is well known that <code>$(\ell^p)^*\cong\ell^q$</code> where <code>$\frac{1}{q}+\frac{1}{p}=1$</code>, and so <code>$$(\ell^p)^{**}=(\ell^q)^*=\ell^p=(\ell^p)\oplus(0).$$</code></p> <p>Note that for $\ell^2$ we have $(\ell^2)^*\cong{\ell^2}$, since $\ell^2$ is a Hilbert space.</p> <p>For <code>$\ell^\infty=\{\{a_n\}: \sup |a_n| \lt \infty\}$</code>, we have <code>$$(\ell^\infty)^* \cong \ell^1\oplus {\rm Null}(C_0),$$</code> and <code>$$(\ell^\infty)^{**}=({\ell}^1)^*\oplus\operatorname{Null}(C_0)^*,$$</code> but $(\ell^1)^\ast=\ell^\infty$, hence <code>$$(\ell^\infty)^{**} = \ell^\infty \oplus{\rm Null} (C_0)^*.$$</code></p> <p>Seeing this pattern, is it true that the double dual of any space $X$ can be written in the form of $X\oplus Y$ for some other space $Y$?</p> http://mathoverflow.net/questions/106695/double-duals-characteristic/106697#106697 Answer by Alex Becker for Double duals characteristic Alex Becker 2012-09-09T00:46:46Z 2012-09-09T00:46:46Z <p>Note that there exists such a $Y$ iff the sequence $$0\to X\overset{\varphi}{\to} (X^*)^*\overset{\eta}{\to} \mathrm{coker}\:\varphi\to 0$$ splits, where $\varphi$ is the canonical injection and $\eta$ the canonical projection. This happens iff there exists an injection $j:\mathrm{coker}\:\varphi\to (X^*)^*$ such that $\varphi(X)\oplus j(\mathrm{coker}\:\varphi)=(X^*)^*$, so can only happen if $\varphi(X)$ is split in $(X^*)^*$. </p> <p>For a counterexample, consider the space $c_0$ of sequences in $\mathbb C$ which converge to $0$. The double dual of $c_0$ is $\ell^\infty$, yet $c_0$ does not split as a subspace of $\ell^\infty$. See for reference page 46 of <a href="http://matematicas.univalle.edu.co/~juancamg/Banach2.pdf" rel="nofollow">this PDF</a>.</p>