We state the Hahn-Banach Theorem. In this session, we look at the simpler algebraic version: Let W be a vector subspace of a vector space V over a field F. Let f be a linear functional from W to F. Then there exists an extension g, a linear functional from V to F. We offer two proofs of this algebraic version. The first one assumes the existence of a basis for any vector space (which is proved using Zorn's lemma). The other proof uses Zorn's lemma directly to prove the existence of an extension.
This will help you understand the way Zorn's lemma is used in the analytic version. We have been painfully slow in this lecture for obvious reasons!
Timestamp provided by Ishwarya and Amit Mittal.
00:00 Disclaimer
0:43 Theme of this lecture series
2:52 Hahn-Banach Theorem
6:50 Proof of the algebraic version of Hahn-Banach Theorem.
15:26 Partially ordered set
18:24 Second proof of the algebraic version by a direct use of Zorn's lemma
22:04 Zorn's Lemma
38:04 Conclusion
