## Hahn-Banach Theorem (real version)

Suppose is a vector space over , has the following properties: and .

Let be a subspace of and a linear functional such that .

Then we can find a linear functional such that and .

**Solution:** Consider the set is a subspace of is a linear functional which extends and on .

Define an order relation on like this if and is an extension for .

We show that in every chain has an upper bound. Suppose is a totally ordered subset of . Then define and if and . This function is well defined, and is a subspace of because the set is totally ordered. Furthermore, from the definition for , we have that . Therefore , and is obviously an upper bound for . By Zorn’s Lemma, we find that has at least one maximal element .

Suppose . Then we can find . Define . Therefore, is a linear subspace in .

Let . Then .

Therefore, we have . Therefore, we can say . Pick one and define , where (unique representation). is linear, and extends on , which means that it extends on . We can check that and is an extension for the maximal element, which is a contradiction. Therefore , and the maximal element is the requested functional