- #1

lmedin02

- 56

- 0

## 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 [itex]{Jx_k}[/itex] in [itex]X''[/itex]. Then for each [itex]x'[/itex] in [itex]X'[/itex] we have that [itex]\sup|Jx_k(x')|[/itex] over all [itex]k[/itex] is finite. I am not sure why this is the case since

[itex]\sup|Jx_k(x')|=\sup|x'(x_k)|\leq\sup||x'||||x_k||[/itex] and [itex]\sup||x_k||[/itex] over all [itex]k[/itex] 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.