# A Banach space is reflexive if a closed subspace and its quotient space are both reflexive

Let $X$ be a Banach space. Let $Y$ be a closed subspace. Suppose that the normed spaces (in fact Banach spaces) $Y$ and $X/Y$ are both reflexive. I need to show that $X$ is reflexive.

I cannot show this but I feel that I could use the fact that a Banach space is reflexive if and only if its closed unit ball is weakly compact. So in this case we know $B_Y$ and $B_{X/Y}$ are weakly compact. Let $\mathcal{U}$ be a weakly open cover for $B_X$. I feel that a finite subcover can be obtained by considering the sets of the form $a+Y\cap B_X\subset X$, which are certainly compact, as $a+Y\cap B_X\subset a+nB_Y\cap B_X$, which is w-closed in the weakly compact set $a+nB_Y$ for some sufficiently large $n$, and the corresponding $a+Y\in X/Y$.

Could anybody suggest anything? Thanks.

• Isn't this equivalent to prove that if $X$ and $Y$ are reflexive, so is $X\times Y$? Commented Jan 16, 2014 at 20:41
• @EmanuelePaolini I'm not sure, but wouldn't it be $X+Y$ Commented Jan 16, 2014 at 20:46
• @EmanuelePaolini You only have an isomorphism $Y\times (X/Y) \cong X$ if $Y$ is a complemented subspace. The premises don't immediately yield that. Commented Jan 16, 2014 at 20:50

A property of a Banach space is a three-space property if whenever $$E$$ is a Banach space, $$F\subseteq E$$ is a closed linear subspace and two of the spaces $$E$$, $$F$$ and $$E/F$$ have the property, then all three of the spaces $$E$$, $$F$$ and $$E/F$$ necessarily have the property.

Reflexivity is a three-space property. For the proof see Theorem 1.11.19 in An Introduction to Banach Space Theory Graduate Texts in Mathematics. Robert E. Megginson.

• Thank you very much for the answer. Thank you for mentioning the three-space property. That is very interesting! I'll read around it. Commented Jan 16, 2014 at 21:48
• You are wellcome! Commented Jan 16, 2014 at 21:48