Summary: Functional Analysis­Math 920 (Spring 2003)
Casim Abbas
April 25, 2003
Contents
1 Preliminary remarks, Notation 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Baire's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Brief review of Integration . . . . . . . . . . . . . . . . . . . . . . 4
2 Normed Linear Spaces 7
2.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Examples of Banach spaces . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Ck
b (), Ck
() and H¨older spaces . . . . . . . . . . . . . . 11
2.2.2 Lp
() and lp
. . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Noncompactness of the unit ball, Uniform Convexity . . . . . . . 50
3 Linear Operators 57

Source: Abbas, Casim - Department of Mathematics, Michigan State University

Collections: Mathematics