Intereting Posts

Subgroups of Prime Power Index
The Uniqueness of a Coset of $R/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R$
Category of profinite groups
Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment
Method of Steepest Descent and Lagrange
Pullback and Pushforward Isomorphism of Sheaves
How to solve congruence $x^y = a \pmod p$?
Uniform convergence problem
A question about sigma-algebras and generators
Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$
How to show that this binomial sum satisfies the Fibonacci relation?
in a topological space only finite subsets are compact sets
Differential equation with separable, probably wrong answer in book
Stone-Čech compactifications and limits of sequences
Calculate the limit $\lim \limits_{n \to \infty} |\sin(\pi \sqrt{n^2+n+1})|$

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.

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- A finite dimensional normed space

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.

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