Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $v$?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.