Originally published at 狗和留美者不得入内. You can comment here or there.
I’m pleased to say that I find the derivation of the Hahn-Banach theorem pretty straightforward by now. Let me first state it, for the real case.
Hahn-Banach theorem: Let 
be a real vector space. Let 
be sublinear. If 
be a linear functional on the subspace 
with 
for 
, then there exists a linear extension of 
to all of (call it 
) such that 
for 
with 
for and 
for all .
To show this, start by taking any 
. We wish to assign some 
to 
that keeps 
as the dominating function in the vector space 
. For this to happen, applying the linearity of and the domination constraint, we can derive
.
This reduces to
.
Such can be proven via
.
Now take the space of linear functionals defined on some specific subspace dominated by . Denote an element of it as 
. We introduce a partial order wherein 
iff 
for and 
. We can apply Zorn’s lemma on this, as we can take the union to derive an upper bound for any chain. Any maximal element is necessarily 
as if the domain is not the entire vector space, we can by above construct a larger element.
