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
.