weak*-closed subspaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:14:39Z http://mathoverflow.net/feeds/question/90396 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90396/weak-closed-subspaces weak*-closed subspaces Denis Poulin 2012-03-06T21:18:08Z 2012-03-07T02:01:23Z <p>Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set $$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$ is a complemented subspace of $X^*$. For example, $c_0$ is a weakly complemented subspace of $l_{\infty}$.</p> <p>Question: Is there a Banach space $X$ such that there is a weak${}^*$-closed subspace $Y$ which is weakly complemented but not complemented in $X$.</p> http://mathoverflow.net/questions/90396/weak-closed-subspaces/90420#90420 Answer by Bill Johnson for weak*-closed subspaces Bill Johnson 2012-03-07T02:01:23Z 2012-03-07T02:01:23Z <p>No. You get <code>$Y^{**}=Y^{\perp\perp}$</code> complemented in <code>$X^{**}$</code> and $Y$, being a dual space, is norm one complemented in $Y^{**}$.</p>