Weak star separable and separable quotient problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:39:40Z http://mathoverflow.net/feeds/question/79073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79073/weak-star-separable-and-separable-quotient-problem Weak star separable and separable quotient problem Qingping Zeng 2011-10-25T13:02:22Z 2011-10-25T15:06:45Z <p>My first question is the following: </p> <p>Q1: Let \$X\$ be a Banach space. If its dual \$X^*\$ is weak* separable, does \$X\$ admit an infinite-dimensional and separable quotient \$X/M\$? </p> <p>To the best of my knowledge, the dual \$X^*\$ is weak* separable, when \$X\$ satisfies one of the following:</p> <p>(i) \$X\$ is separable;</p> <p>(ii) \$X\$ is the dual of a separable Banach space;</p> <p>(iii) \$X\$ is Hereditary Indecomposable Banach space. That is, every infinite-dimensional closed subspace of \$X\$ can not be written as a direct sum of two infinite-dimensional closed subspaces. </p> <p>And, I see that if \$X\$ satisfies (i) or (ii), \$X\$ admit an infinite-dimensional and separable quotient. </p> <p>Q2: Is it true that \$X\$ admit an infinite-dimensional and separable quotient, if \$X\$ satisfies (iii) ? </p> http://mathoverflow.net/questions/79073/weak-star-separable-and-separable-quotient-problem/79088#79088 Answer by Bill Johnson for Weak star separable and separable quotient problem Bill Johnson 2011-10-25T15:06:45Z 2011-10-25T15:06:45Z <p>Q1 is equivalent to the separable quotient problem. Indeed, given \$Y\$ infinite dimensional, let \$W\$ be any separable infinite dimensional subspace of <code>\$X^*\$</code> and let \$Y\$ be the annihilator of \$W\$ in \$X\$. Then the dual of \$X/Y\$ is the weak<code>\$^*\$</code> closure of \$W\$ in <code>\$X^*\$</code>. </p>