Functional Analysis, spring 2005, J. Hulshof

Credits: 6 ECTS

Examination: homework and oral exam.

Week 14-21. Hours: Tuesdays, 13.30-16.15 (F654), Thursdays 13.30-16.15 (F640).

Practical points:

Thursdays session will include exercise class.

Examination: combination of home work and oral exam.

Planning:

Week 1. Basic theory of Banach spaces. Finite-dimensional spaces. Spaces of continuous functions.

Banach fixed point theorem and application to o.d.e.'s. Equicontinuity and Ascoli-Arzela.

Hand in homework Ex. 8 and 9.

Ex 8 has a misprint. The functions indexed by a should be defined by
|x-a| to the power gamma.

Ex 9 has a misprint, gamma should be larger than one.

Week2. Sequence spaces consisting of points with countably many coordinates, the so-called small l p spaces.
Hilbert spaces.

Exercise class thursday: Ex 11, 12 (not so easy, you have to construct one), 1.8, 1.9, 1.11 (all for little lp), 1.12, 1.13, 1.15.

Hand in home work: Ex 11, 1.9.

Week 3. Hilbert spaces. Simple spectacular theorems, including a generalisation of
the theorem about diagonalisation of symmetric matrices.

Exercise class on Thursday: 13, 3.1, 3.2, 3.3, 3.6.

Hand in homework 1.8.

Week 4. Application to boundary value problems, see new subsection 2.2. of Additional material.
Continuous linear functions on Banach spaces. Complete characterisation in sequence spaces and Hilbert spaces.
Discussed Lebesque spaces of measurable integrable functions. (Course on Tuesday and Thursday)

Week 5.
No session on Tuesday, exercise class on Thursday. Section 2.2. Possibly also 3.12, but with some explanation of Vincent.

Week 6. Integral operators, linear maps with finite-dimensional range. Course on Tuesday and Wednesday (in R223) afternoon.

On Wednesday I discussed Section 3.3 and Prop. 3.13 in relation to Problem 3.13 with p=1 and q=0.

Thursday exercise class. Remaining programme: Section 2.2 and Section 3 of the notes, plus ex 3.8 in the book.

Week 7. Exercise class on Tuesday and Thursday. Hand in home work: 2.2 (viii),(x); 3 17,18,19. Ex 3.8.

TRY TO FINISH THE HOME WORK THIS WEEK!

Prerequisites: a general background in analysis and linear algebra.

Literature:
Infinite-Dimensional Dynamical Systems:

An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors

(Cambridge Texts in Applied Mathematics)

James C. Robinson

ISBN: 0521635640

available from STORM.

This is the book used in the national master programme for the courses Partial Differential Equations
and Infinite Dimensional Dynamical Systems. Part I of this book deals with functional analysis.
As is clear from the title of the book, we will treat functional analysis as a tool for solving equations. The
material in the book has some overlap with that in the book of Schechter, which is used
the Functional Analysis course of the same national master programme, but not too much.