Course plan: We will be learning Functional Analysis.
Assignments, additional related material will be posted on this page.
Course evaluation, policy details and dates: see here
Resources: For a brief historical account by Neal L. Carothers : see here.
Lecture plan:
2nd Jan 2026 : Recall about Metric spaces, Examples of metrics d_p on K^n, inequalities by Young, Hölder and Minkowski
5th Jan : Lecture 1: Further examples of metric spaces: sequences spaces, function spaces
7th Jan : Lecture 2: Normed spaces, basic properties of normed spaces
9th Jan : Lecture 3: Subspaces of normed spaces, Quotients of normed spaces
12th Jan : Lecture 4: Riesz' lemma, Finite dimensional subspaces of normed spaces are complete.
14th Jan : Lecture 5: Topological characterization of Finite dimesional normed spaces, Equivalence of norms
19 th Jan : Lecture 6: Continuity of linear maps
20st Jan : L7: Characterization of countinuous maps
21st Jan : L8: Bounded linear maps, norm of operator
23rd Jan : L9: Hahn-Banach theorems
30 Jan: No class. Quiz 1 this week on Friday 30th Jan
2nd Feb: L10: Hahn-Banach theorems, Applications
3rd Feb: L11: Banach spaces, Examples, Summability
4th Feb: L12: New Banach spaces out of old ones, Completion of a normed space
9th Feb : L13: Uniform boundedness principles
11th Feb: L14: Open map theorem
13th Feb: L15: Closed graph theorem
16th Feb: L16: Examples of failure of CGT and OMT, Applications of CGT and OMT
18-21 Feb: Minor exam period
23 Feb: Problems from Minor exam
25 Feb: Bounded linear functionals, dual spaces,
2nd March: Dual of sequence spaces l^p
6th March: reflexive normed spaces
9th March: reflexivity versus separability, Measurable sets/functions
11th March: Measurable sets/functions ctd, L^p spaces
17th March : L^p-spaces continued
18th March : (Geometry of Hilbert spaces and applications) Inner product spaces, Examples
20th March : Hilbert spaces, examples, completion
23rd March : Orthonormality
25 March :
27 March :
30 March :
1 April :
6 April :
8 April :
10 April :
13 April :
15 April :
17 April :
20 April :