# Closedness of the image of the closed unit ball under a linear operator from a reflexive Banach space to an arbitrary Banach space

Let $V$ be a reflexive banach space. If $W$ is a Banach space and if $T$ is in $L(V,W)$, show that $T(B)$ is closed in $W$ where $B$ is closed unit ball in $V$, the problem is in the chapter of weak and weak$^{\ast}$ topologies but I am not getting any hint what result I should use. please help.

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You wrote that $T$ is a reflexive Banach space, but $T$ is the operator. Is it $V$ or $W$ that should be reflexive? Also, your questions will be easier to read and get more attention if you use complete sentences and correct grammar and punctuation. – Nate Eldredge Apr 13 '11 at 13:13

I'll assume you mean that $V$ is reflexive, since that's the only way that makes sense.
1. Since $V$ is reflexive, what can you say about the unit ball of $V$ (in a certain topology...)?
2. To show $T(B)$ is closed, suppose $v_n$ in $B$ with $Tv_n \to w$. You need to produce $v \in B$ with $Tv = w$. In light of the previous hint, can you think of a candidate for $v$?