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 .
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