Functional Analysis (Spring 2018)

I strongly suggest the students refer to this course page regularly. For pre-requisite, one should read the item 01 from the cabinet.

Lectures

• 1. 08/01 Motivation, Normed Linear Spaces (NLS), Example of a discontinuous linear operator.

1. 09/01 L^p spaces, Minkowski's inequality. (118-120 [4])

   1. 10/01 Hölder's inequality, A necessary and sufficient condition for the completeness of an NLS. (121-124 [4])

   2. 13/01 Completeness of L^p for p finite. Examples of incomplete NLSs. (125-126 [4])

   3. 15/01 Littlewood's 1st and 2nd principle. Set of step functions on [a,b] is dense in L^p([a,b]).

   4. 16/01 L^p approximation of a given L^p function by a simple function, continuous function, and step function. (127-129 [4])

   5. 17/01 L^q spaces where 1/q = 1-1/p. Lemmata related to Riesz Representation Theorem. (130-131, 282-284 [4])

   6. 22/01 Riesz representation theorem for L^p spaces for finite p≥1. (284-287 [4])

   7. 23/01 Subspaces of an abstract normed linear space, closed subspace, examples of closed subspaces, Quotient space, Riesz Lemma. (68-71 [1])

2. 24/01 Finite dimensional (fd) space: An fd subspace is closed. A sequence of points in an fd space converges iff the coordinates converge. An NLS is fd iff the closed unit ball is compact. (72-75 [1])

3. 29/01 Properties of vector addition of open or convex sets. Hahn Banach separation Theorem. (104-110 [1])

4. 30/01 Tutorial

5. 31/01 Quiz 1

6. 05/02 Properties of bounded linear operators. (84-87 [1])

7. 06/02 Definition of operator norm and its alternative expressions, Spaces of bounded linear operators BL(X,Y), and necessary & sufficient condition for completeness. (94-98, 127-129[1], 220 [4])

8. 07/02 Hahn-Banach extension Theorem. (221-224 [4])

9. 12/02 Application of Hahn-Banach extension Theorem. (225-226 [4])

10. 13/02 Canonical embedding of an NLS X into X'', Reflexivity. (227-228 [4])

11. 17/02 Quiz 2 (11:00 -11:30) in LHC 304

12. 27/02 Midsem (15:00-17:00)

13. 28/02 Proof of a lemma, Open mapping theorem. ( 229 - 230 [4])

• 1. 05/03 Proof of Closed Graph Theorem as an application of open mapping theorem, Uniform boundedness principle. (231 - 232 [4])

14. 06/03 Complete orthonormal system in a separable Hilbert space. (246-247 [4])

15. 07/03 Riesz representation theorem for separable Hilbert space. (248-250 [4])

• 1. 12/03 Necessary and sufficient condition for continuity of a projection operator (169-170 [1]), Examples: X=C([0,1]) with L¹ norm, and P(x):=x(0)1_{[0,1]} is a discontinuous projection.

16. 13/03 Topological vector space (233-234 [4]), Weak and Weak* topology and Alaoglu Theorem. (236-238 [4])

17. 14/03 Quiz 3

18. 19/03 Necessary and sufficient condition for invertibility of bounded linear operator. (192-195 [1])

19. 20/03 Definition of eigenspectra, approximate spectra, and spectra of an operator. Spectral properties of a finite rank bounded linear operator on an NLS. (196-197 [1])

20. 21/03 The domain and the continuity property of inversion map for bounded linear operators on a Banach space. (198-202 [1])

21. 26/03 Definition of the compact linear operators. (302-303 [1])

22. 27/03 Properties of the compact linear operators. (303-305 [1])

23. 28/03 Weakly convergent sequence is bounded (262 [1]), A bounded sequence in reflexive space has a weakly convergent sub-sequence (288 [1]), Compact Image of a weakly convergent sequence is convergent. (312 [1])

24. 02/04 Tutorial & Quiz 4

25. 03/04 Every nonzero spectral value of a compact linear operator is an eigenvalue. (317-321 [1])

26. 04/04 Remaining part of the previous lecture.

27. 09/04 The spectrum of a compact linear operator is at most countable (322-324 [1]), Computing spectrum of an example (204-207 [1]).

28. 10/04 Transpose of a bounded linear operator (222 [1]), Adjoint of a bounded linear operator on a Hilbert space (448-449 [1]),

29. 11/04 Norm of transpose (225 [1]). normal operator, properties of a self-adjoint operator (448-465 [1]),

30. 16/04 Spectral theory of compact self-adjoint operator (505-511 [1]).

31. 17/04 Revision

32. 23/04 Endsem (LHC 201) 3:00-5:00 PM