文档介绍：)XQFWLRQDO $QDO\VLV
D
qpv Trp
hy Uur
v Cvyir
Thpr Tp
v
/HFWXUH LQ :LQWHUVHPHVWHU
Grp
r
) Q
s 9
Srir
tr
X
vr q i Uuh Arur
Srvrq i Xystht ***@vqr
Functional Analysis Page 1
List of contents
1. Hilbert spaces 3
Basic definitions and results 3
Orthogonality and orthonormal bases 8
Isomorphisms 16
2. Bounded linear operators 18
Bounded linear mappings 18
Adjoint operators 25
Projection operators 30
Baire’s Category Theorem and Banach-Steinhaus-Theorem 34
3. Spectral analysis of bounded linear operators 36
The order relation for bounded selfadjoint operators 36
operators on a Hilbert space 43
Eigenvalues pact operators 51
The spectral position of pact linear operator 58
Functional Analysis Page 2
Introduction to Spectral Theory in Hilbert Space
The aim of this course is to give a very modest introduction to the extremely rich and well-
developed theory of Hilbert spaces, an introduction that hopefully will provide the students
with a knowledge of some of the fundamental results of the theory and will make them
familiar with everything needed in order to understand, believe and apply the spectral theorem
for selfadjoint operators in Hilbert space. This implies that the course will have to give
answers to such questions as
- What is a Hilbert space?
- What is a bounded operator in Hilbert space?
- What is a selfadjoint operator in Hilbert space?
- What is the spectrum of such an operator?
- What is meant by a spectral position of such an operator?
LITERATURE:
- English:
• G. Helmberg: Introduction to Spectral Theory in Hilbert space
(North-Holland p., Amsterdam-London)
• R. Larsen: Functional Analysis, an introduction
(Marcel Dekker Inc., New York)
• M. Reed and B. Simon: Methods of Modern Mathematical Physics I:
Functional Analysis
(Academic Press, New York-London)
- German:
• H. Heuser: Funktionalanalysis, Theorie