When $L\left(E^{*},F^{**}\right)$= $L\left(E^{*},F\right)$+ the adjoints ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:31:23Z http://mathoverflow.net/feeds/question/23300 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23300/when-l-lefte-f-right-l-lefte-f-right-the-adjoints When $L\left(E^{*},F^{**}\right)$= $L\left(E^{*},F\right)$+ the adjoints ? Ady 2010-05-02T23:58:47Z 2010-05-02T23:58:47Z <p>This question is somehow related to <a href="http://mathoverflow.net/questions/22799" rel="nofollow">http://mathoverflow.net/questions/22799</a>. Let $E$ and $F$ be two (say real) Banach spaces, and let </p> <p>$\mathcal{P}_{E,F}$ = "<em>For every linear continuous operator</em> $T$ <em>from</em> $E^{*}$ <em>to the bidual of</em> $F$ <em>there exist some</em> $S\in L(E^{*},F)$ <em>and some</em> $Q\in L(F^{*},E)$ <em>such that</em> $T=Q^{*} +S$ ". Looking at <a href="http://mathoverflow.net/questions/23221" rel="nofollow">http://mathoverflow.net/questions/23221</a>, it follows that $\mathcal{P}_{E,F}$ </p> <p>is false if $E=\ell^{1}\left(\Gamma\right)$, and $F=c_{0}\left(\mathbb{N},H\right)$, where $H$ is a Hilbert space, and both $\mid\Gamma\mid$and $dens$ $H$ are big enough. My [maybe rather vague] question would be: are there some known (and non-trivial) sufficient conditions on the pair $\left(E,F\right)$ s.t. $\mathcal{P}_{E,F}$ is true ? Thx in advance.</p>