Theorem 11 (Lax Functional Analysis): A weak* convergent sequence of points in a Banach space is uniformly bounded.
We will need a previous Theorem 3: is a Banach space, a collection of bounded linear functionals such that at every point of , for all . Then there is a constant such that for all .
Sketch of proof:
Weak* convergence means , thus there exists such that for all , we have , which in turns means via the triangle inequality. We have managed to bound the terms greater than equals to .
For those terms less than , we have .
Thus, we may take . The crucial thing is that depends only on , not .
Then, use Theorem 3, we can conclude that for all .