We look at two simple cases of continuous extension: (i) When Y is dense in X and (ii) when Y is a closed vector subspace of a Hilbert space X. We then explore how to extend to Y+Rx where x is in X but not in Y.
Timestamp provided by Amit Mittal.
00:00: Introduction
00:43 : Overview of the lecture
01:16: Recalling Hahn-Banach Theorem
02:16: First simplest case , when Y is dense in X
10:00: Second simplest case , when Y is a closed vector subspace of a Hilbert space X
11:28: Riesz Representation Theorem
14:25: Summary
15:25 : General case , Extend linear functional f to g such that operator norm of g coinciding with the operator norm of f.
24:50 : Third case , extend to Y+Rx where x is in X but not in Y. Most crucial step.
32:33 : Thank you for watching
