Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $H$ is a complex Hilbert space and $$w: \mathcal{B}(H) \longrightarrow \mathbb{C}$$ is a bounded linear functional on $\mathcal{B}(H)$ such that $w$ is continuous even if $\mathcal{B}(H)$ is given the strong operator topology. I am supposed to prove that there exists $c>0$ and $h_1, \dots, h_n \in H$ such that for all $x \in \mathcal{B}(H)$, one has $$|w(x)| \leq c \sqrt{\sum_{i = 1}^n \|x(h_i)\|^2}.$$ I have no idea how to begin. To me, the strong operator topology is given by the family of seminorms $$\{\|x\|_h = \|x(h)\| : h \in H\}.$$

share|improve this question
1  
Hint: What do the open sets of the strong operator topology look like (specifically, what is a basis for this topology)? What does this say about $\{x : |w(x)| < 1\}$? –  Nate Eldredge May 5 '12 at 5:09
    
Thanks, this was very helpful. –  Jeff May 5 '12 at 6:24
    
@NateEldredge You could post your comment as answer. –  Davide Giraudo May 5 '12 at 9:37

1 Answer 1

up vote 3 down vote accepted

Hint: What do the open sets of the strong operator topology look like (specifically, what is a basis for this topology)? What does this say about $\{x:|w(x)|<1\}$?

share|improve this answer

Your Answer

 
discard

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.