Intereting Posts

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Find the Remainder when $792379237923$…upto 400 digits is divided by $101$?
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How original RS codes and the corresponding BCH codes are related?
A non-negative matrix has a non-negative inverse. What other properties does it have?
Vector Spaces: Redundant Axiom?
Solving an integral with substitution method
Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.
Complex finite dimensional irreducible representation of abelian group
Approximating a $\sigma$-algebra by a generating algebra
The action of SU(2) on the Riemann sphere
Getting Students to Not Fear Confusion

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 function that is $L^p$ for all $p$ but is not $L^\infty$?

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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