Take the 2-minute tour ×  I want to prove:  If $X$ is reflexive,
$T\in B(X,Y)$ and $\| Tx_n - Tx \| \rightarrow 0$ whenever $x_n
\rightarrow x$ weakly then $T$ is compact.  Here is my attempt
at the proof: Assuming all the conditions hold, we'll want to
show $T(ball X )$ has compact closure, or equivalently, any
sequence in $T(ball X)$ has convergent subsequence, where
$ballX$ is the unit ball of $X$.  So, let $(x_n)$ be a net in
$ballX$ which converges weakly to $x$ then $Tx_n$ converges
(strongly) to $Tx$  (?) then $(x_n)$ has a convergent
subsequence, then $T(ballX)$ has compact closure.  This doesn't
sound right... and I don't know where to use the fact that $X$
is reflexive.  share|improve this question A bounded sequence in
$X$ has a weakly convergent subsequence (this is where you use
reflexivity). So, by hypothesis, every bounded sequence in $X$
has a subsequence whose image under $T$ is norm convergent. – 
David Mitra Jun 5 '12 at 0:31 add comment   Your Answer
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