Weakly convergent sequences are bounded

[lmedin02]

Homework Statement

I would like to show that a weakly convergent sequence is necessarily bounded.

The Attempt at a Solution

I would like to conclude that if I consider a sequence ${Jx_k}$ in $X''$. Then for each $x'$ in $X'$ we have that $\sup|Jx_k(x')|$ over all $k$ is finite. I am not sure why this is the case since
$\sup|Jx_k(x')|=\sup|x'(x_k)|\leq\sup||x'||||x_k||$ and $\sup||x_k||$ over all $k$ does not seem finite.

From here I would be able to make my conclusion by applying the uniform boundedness principle or the Banach-Steinhaus Theorem.

 

[morphism]

I don't really understand your notation.

Shouldn't you just look at various functionals applied to your sequence - since each of these will converge, you get pointwise boundedness. But then apply Hahn-Banach to find a functional that attains the norm of your sequence at a specific point. So...

 

[lmedin02]

Thanks for your comments. I was able to resolve my issue with this problem.