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泛函分析(剑桥大学英文原版,NICE)
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Functional Analysis
Dr A. J. Wassermann
1
Lent term 1999
1
L
A
T
E
Xed by Tim Perutz { comments to
soc-archim-notes@lists.cam.ac.uk
.
Contents
1 Intro duction 4
2 Metric spaces 6
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Compactness and completeness . . . . . . . . . . . . . . . . . 9
3 Normed spaces 13
3.1 Continuity prop erties of the algebraic op erations . . . . . . . . 14
3.2 Completeness . . . . . . . . . . . . . . . .
. . . . . . . . . . . 14
3.3 The uniform norm . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 The space
`
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 The
`
p
spaces . . . . . . . . . . . . . . . . . . . . . .
. . . . . 16
3.6 The
L
p
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Equivalentnorms . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.8 Bounded operators and linear functionals . . . . . . . . . . . . 21
4 Inner pro duct spaces 25
4.1 Closest points and the pro jection theorem . . . . . . . . . . . 26
4.2 Orthonormal bases and Parseval's equation . . . . . . . . . . . 30
4.3 Orthonormal bases and the Gram-Schmidt pro cess . . . . . . . 32
4.4 The dual of a Hilbert space:
the Riesz representation theorem . . . . . . . . . . . . . . . . 33
5 Fourier series and the Dirichlet problem 36
5.1 The Dirichlet problem . . . . . . . . . . . . . . . . . . . .
. . 40
5.2 Uniformly convergent Fourier series . . . . . . . . . . . . . . . 42
6 Application to theta, gamma and zeta functions 45
6.1 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 The gamma and beta functions . . . . . . . . . . . . . . . . . 47
6.3 The Riemann zeta function . . . . . . . . .
. . . . . . . . . . 48
2
CONTENTS
3
7 The Stone-Weierstrass Theorem 50
7.1 Imp ortant example: Hermite functions . . . . . . . . . . . . . 57
7.2 Pro ducts and easy `Fubini' theorem . . . . . . . . . . . . . . . 58
Note.
The nal chapter of the course, on Leb esgue integration, is omitted
here as this was supplied in printed form byDrWassermann. These notes do
not always repro duce verbatim what Dr Wassermann wrote on the b oard, but
the changes are minor - they suited me, and they might not please everyone.
I would also encourage you
not
to see these notes as a substitute for going
to the lectures, which contain more jokes and are billed by the lecturer as
`mellow'...
T.P.
Chapter 1
Intro duction
ther electronic lecture notes are available from the Archimedeans. A list of
available courses, and the notes may be downloaded from
http://www.cam.ac.uk/CambUniv/Soc ieties/arch im/notes.htm
or you can email
soc-archim-notes@lists.cam.ac.uk
to get a copyofthe
sets you require.
4
Copyright (c) The Archimedeans, Cambridge University.
All rights reserved.
Redistribution and use of these notes in electronic or printed form, with
or without mo dication, are p ermitted provided that the following conditions
are met:
1. Redistributions of the electronic les must retain the ab ove copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in printed form must reproduce the ab ove copyright
notice, this list of conditions and the following disclaimer.
3. All materials derived from these notes must display the following ac-
knowledgement:
This pro duct includes notes
develop ed by The Archim-
edeans, Cambridge University and their contributors.
4. Neither the name of The Archimedeans nor the names of their contrib-
utors may be used to endorse or promote products derived from these
notes.
5. Neither these notes nor any derived pro ducts may b e sold on a for-prot
basis, although a fee mayberequired for the physical act of copying.
6. You must cause any edited versions to carry prominent notices stating
that you edited them and the date of any change.
THESE NOTES ARE PROVIDED BY THE ARCHIMEDEANS AND
CONTRIBUTORS \AS IS" AND ANY EXPRESS OR IMPLIED WAR-
RANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WAR-
RANTIES OF MERCHANTABILITY AND FITNESS FOR A PAR
TICU-
LAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ARCHIM-
EDEANS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDI-
RECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES HOWEVER CAUSED AND ON ANY THEORY OF LIABIL-
ITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (IN-
CLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
OUT OF THE USE OF THESE NOTES, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
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