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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1995
|
| Edition: | Corr. 2. print. |
| Series: | Graduate texts in mathematics
118 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 277 S. |
| ISBN: | 3540967885 0387967885 |
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MARC
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| 245 | 1 | 0 | |a Analysis now |c Gert K. Pedersen |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1995 | |
| 300 | |a XIV, 277 S. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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|---|---|
| adam_text |
CONTENTS
PREFACE
VII
C
HAPTER
1
1
GENERAL
TOPOLOGY
1.1.
ORDERED
SETS
1
THE
AXIOM
OF
CHOICE,
ZORN
'
S
LEMMA,
AND
CANTORS
'
S
WELL-ORDERING
PRINCIPLE;
AND
THEIR
EQUIVALENCE.
EXERCISES.
1.2.
TOPOLOGY
8
OPEN
AND
CLOSED
SETS.
INTERIOR
POINTS
AND
BOUNDARY.
BASIS
AND
SUBBASIS
FOR
A
TOPOLOGY.
COUNTABILITY
AXIOMS.
EXERCISES.
1.3.
CONVERGENCE
13
NETS
AND
SUBNETS.
CONVERGENCE
OF
NETS.
ACCUMULATION
POINTS.
UNIVERSAL
NETS.
EXERCISES.
1.4.
CONTINUITY
17
CONTINUOUS
FUNCTIONS.
OPEN
MAPS
AND
HOMEOMORPHISMS.
INITIAL
TOPOLOGY.
PRODUCT
TOPOLOGY.
FINAL
TOPOLOGY.
QUOTIENT
TOPOLOGY.
EXERCISES.
1.5.
SEPARATION
23
HAUSDORFF
SPACES.
NORMAL
SPACES.
URYSOHN
'
S
LEMMA.
TIETZE
'
S
EXTENSION
THEOREM.
SEMICONTINUITY.
EXERCISES.
1.6.
COMPACTNESS
30
EQUIVALENT
CONDITIONS
FOR
COMPACTNESS.
NORMALITY
OF
COMPACT
HAUSDORFF
SPACES.
IMAGES
OF
COMPACT
SETS.
TYCHONOFF
'
S
THEOREM.
COMPACT
SUBSETS
OF
R".
THE
TYCHONOFF
CUBE
AND
METRIZATION.
EXERCISES.
XII
CONTENTS
1.7.
LOCAL
COMPACTNESS
36
ONE-POINT
COMPACTIFICATION.
CONTINUOUS
FUNCTIONS
VANISHING
AT
INFINITY.
NOR
MALITY
OF
LOCALLY
COMPACT,
A-COMPACT
SPACES.
PARACOMPACTNESS.
PARTITION
OF
UNITY.
EXERCISES.
C
HAPTER
2
43
BANACH
SPACES
2.1.
NORMED
SPACES
43
NORMED
SPACES.
BOUNDED
OPERATORS.
QUOTIENT
NORM.
FINITE-DIMENSIONAL
SPACES.
COMPLETION.
EXAMPLES.
SUM
AND
PRODUCT
OF
NORMED
SPACES.
EXERCISES.
2.2.
CATEGORY
52
THE
BAIRE
CATEGORY
THEOREM.
THE
OPEN
MAPPING
THEOREM.
THE
CLOSED
GRAPH
THEOREM.
THE
PRINCIPLE
OF
UNIFORM
BOUNDEDNESS.
EXERCISES.
2.3.
DUAL
SPACES
56
THE
HAHN-BANACH
EXTENSION
THEOREM.
SPACES
IN
DUALITY.
ADJOINT
OPERATOR.
EXERCISES.
2.4.
SEMINORMED
SPACES
62
TOPOLOGICAL
VECTOR
SPACES.
SEMINORMED
SPACES.
CONTINUOUS
FUNCTIONALS.
THE
HAHN-BANACH
SEPARATION
THEOREM.
THE
WEAK*
TOPOLOGY.
W*-CLOSED
SUBSPACES
,
AND
THEIR DUALITY
THEORY.
EXERCISES.
2.5.
W*-COMPACTNESS
69
ALAOGLU
'
S
THEOREM.
KREIN-MILMAN
'
S
THEOREM.
EXAMPLES
OF
EXTREMAL
SETS.
EXTRE
MAL
PROBABILITY
MEASURES.
KREIN-SMULIAN
'
S
THEOREM.
VECTOR-VALUED
INTEGRATION.
EXERCISES.
C
HAPTER
3
79
HILBERT
SPACES
3.1.
INNER
PRODUCTS
79
SESQUILINEAR
FORMS
AND
INNER
PRODUCTS.
POLARIZATION
IDENTITIES
AND
THE
CAUCHY
SCHWARZ
INEQUALITY.
PARALLELLOGRAM
LAW.
ORTHOGONAL
SUM.
ORTHOGONAL
COMPLE
MENT.
CONJUGATE
SELF-DUALITY
OF
HILBERT
SPACES.
WEAK
TOPOLOGY.
ORTHONORMAL
BASIS.
ORTHONORMALIZATION.
ISOMORPHISM
OF
HILBERT
SPACES.
EXERCISES.
3.2.
OPERATORS
ON
HILBERT
SPACE
88
THE
CORRESPONDENCE
BETWEEN
SESQUILINEAR
FORMS
AND
OPERATORS.
ADJOINT
OPERATOR
AND
INVOLUTION
IN
B(§).
INVERTIBILITY,
NORMALITY,
AND
POSITIVITY
IN
B(5).
THE
SQUARE
ROOT.
PROJECTIONS
AND
DIAGONALIZABLE
OPERATORS.
UNITARY
OPERATORS
AND
PARTIAL
ISOMETRIES.
POLAR
DECOMPOSITION.
THE
RUSSO-DYE-GARDNER
THEOREM.
NUMERICAL
RADIUS.
EXERCISES.
3.3.
COMPACT
OPERATORS
105
EQUIVALENT
CHARACTERIZATIONS
OF
COMPACT
OPERATORS. THE
SPECTRAL THEOREM
FOR
NORMAL,
COMPACT
OPERATORS.
ATKINSON
'
S
THEOREM.
FREDHOLM
OPERATORS
AND
INDEX.
INVARIANCE
PROPERTIES
OF
THE
INDEX.
EXERCISES.
CONTENTS
3.4.
THE
TRACE
DEFINITION
AND
INVARIANCE
PROPERTIES
OF
THE
TRACE.
THE
TRACE
CLASS
OPERATORS
AND
THE
HILBERT-SCHMIDT
OPERATORS.
THE
DUALITIES
BETWEEN
B
1
(F )
AND
B(§).
FREDHOLM EQUATIONS.
THE
STURM-LIOUVILLE
PROBLEM.
EXERCISES.
C
HAPTER
4
SPECTRAL
THEORY
4.1.
BANACH
ALGEBRAS
IDEALS
AND
QUOTIENTS.
UNIT
AND
APPROXIMATE
UNITS.
INVERTIBLE
ELEMENTS.
C.
NEUMANN
'
S
SERIES.
SPECTRUM
AND
SPECTRAL
RADIUS.
THE
SPECTRAL
RADIUS
FORMULA.
MAZUR
'
S
THEOREM.
EXERCISES.
4.2.
THE
GELFAND
TRANSFORM
CHARACTERS
AND
MAXIMAL
IDEALS.
THE
GELFAND
TRANSFORM.
EXAMPLES,
INCLUDING
FOURIER
TRANSFORMS.
EXERCISES.
4.3.
FUNCTION
ALGEBRAS
THE
STONE-WEIERSTRASS
THEOREM.
INVOLUTION
IN
BANACH
ALGEBRAS.
C*-ALGEBRAS.
THE
CHARACTERIZATION
OF
COMMUTATIVE
C*-ALGEBRAS.
STONE-CECH
COMPACTIFICATION
OF
TYCHONOFF
SPACES.
EXERCISES.
4.4.
THE
SPECTRAL
THEOREM,
I
SPECTRAL
THEORY
WITH
CONTINUOUS
FUNCTION
CALCULUS.
SPECTRUM
VERSUS
EIGENVALUES.
SQUARE
ROOT
OF
A
POSITIVE
OPERATOR.
THE
ABSOLUTE
VALUE
OF
AN
OPERATOR.
POSITIVE
AND
NEGATIVE
PARTS
OF
A
SELF-ADJOINT
OPERATOR.
FUGLEDE
'
S
THEOREM.
REGULAR
EQUIVALENCE
OF
NORMAL
OPERATORS.
EXERCISES.
4.5.
THE
SPECTRAL
THEOREM,
II
SPECTRAL
THEORY
WITH
BOREL
FUNCTION
CALCULUS.
SPECTRAL
MEASURES.
SPECTRAL
PROJECTIONS
AND
EIGENVALUES.
EXERCISES.
4.6.
OPERATOR
ALGEBRA
STRONG
AND
WEAK
TOPOLOGY
ON
B($).
CHARACTERIZATION
OF
STRONGLY/WEAKLY
CONTIN
UOUS
FUNCTIONALS.
THE DOUBLE
COMMUTANT
THEOREM.
VON
NEUMANN
ALGEBRAS.
THE
R-WEAK
TOPOLOGY.
THE
A-WEAKLY
CONTINUOUS
FUNCTIONALS.
THE
PREDUAL
OF
A
VON
NEUMANN
ALGEBRA.
EXERCISES.
4.7.
MAXIMAL
COMMUTATIVE
ALGEBRAS
THE
CONDITION
91
=
91'.
CYCLIC
AND
SEPARATING
VECTORS.
AS
MULTIPLICATION
OPERATORS.
A
MEASURE-THEORETIC
MODEL
FOR
MAQA
'
S.
MULTIPLICITY-FREE
OPERATORS.
MA^A
'
S
AS
A
GENERALIZATION
OF
ORTHONORMAL
BASES.
THE
SPECTRAL
THEOREM
REVISITED.
EXERCISES.
C
HAPTER
5
UNBOUNDED
OPERATORS
5.1.
DOMAINS,
EXTENSIONS,
AND
GRAPHS
DENSELY
DEFINED
OPERATORS.
THE
ADJOINT
OPERATOR.
SYMMETRIC
AND
SELF-ADJOINT
OPERATORS.
THE
OPERATOR
T*T.
SEMIBOUNDED
OPERATORS.
THE
FRIEDRICHS
EXTENSION.
EXAMPLES.
XIII
115
127
128
137
144
156
162
171
180
191
192
XIV
CONTENTS
5.2.
THE
CAYLEY
TRANSFORM
203
THE
CAYLEY
TRANSFORM
OF
A
SYMMETRIC
OPERATOR.
THE
INVERSE
TRANSFORMATION.
DEFECT
INDICES.
AFFILIATED
OPERATORS.
SPECTRUM
OF
UNBOUNDED
OPERATORS.
5.3.
UNLIMITED
SPECTRAL
THEORY
209
NORMAL
OPERATORS
AFFILIATED
WITH
A
MAA.
THE
MULTIPLICITY-FREE
CASE.
THE
SPECTRAL
THEOREM
FOR
AN
UNBOUNDED,
SELF-ADJOINT
OPERATOR.
STONE
'
S
THEOREM.
THE
POLAR
DECOMPOSITION.
C
HAPTER
6
221
INTEGRATION
THEORY
6.1.
RADON
INTEGRALS
221
UPPER
AND
LOWER
INTEGRAL.
DANIELL
'
S
EXTENSION
THEOREM.
THE
VECTOR
LATTICE
-$?'(.).
LEBESGUE
'
S
THEOREMS
ON
MONOTONE
AND
DOMINATED
CONVERGENCE.
STIELTJES
INTEGRALS.
6.2.
MEASURABILITY
228
SEQUENTIALLY
COMPLETE
FUNCTION
CLASSES,
A-RINGS
AND
A-ALGEBRAS.
BOREL
SETS
AND
FUNCTIONS.
MEASURABLE
SETS
AND
FUNCTIONS.
INTEGRABILITY
OF
MEASURABLE
FUNCTIONS.
6.3.
MEASURES
235
RADON
MEASURES.
INNER
AND
OUTER
REGULARITY.
THE
RIESZ
REPRESENTATION
THEOREM.
ESSENTIAL
INTEGRAL.
THE
A-COMPACT
CASE.
EXTENDED
INTEGRABILITY.
6.4.
L'
'
-SPACES
239
NULL
FUNCTIONS
AND
THE
ALMOST
EVERYWHERE
TERMINOLOGY.
THE
HOLDER
AND
MINKOWSKI
INEQUALITIES.
EGOROFF
'
S
THEOREM.
LUSIN
'
S
THEOREM.
THE
RIESZ-FISCHER
THEOREM.
APPROXIMATION
BY
CONTINUOUS
FUNCTIONS.
COMPLEX
SPACES.
INTERPOLA
TION
BETWEEN
^"
'
-SPACES.
6.5.
DUALITY
THEORY
247
7-COMPACTNESS
AND
A-FINITENESS.
ABSOLUTE
CONTINUITY.
THE
RADON-NIKODYM
THEOREM.
RADON
CHARGES.
TOTAL
VARIATION.
THE
JORDAN
DECOMPOSITION.
THE
DUALITY
BETWEEN
//-SPACES.
6.6.
PRODUCT
INTEGRALS
255
PRODUCT
INTEGRAL.
FUBINI
'
S
THEOREM.
TONELLI
'
S
THEOREM.
LOCALLY
COMPACT
GROUPS.
UNIQUENESS
OF
THE
HAAR
INTEGRAL.
THE
MODULAR
FUNCTION.
THE
CONVOLUTION
ALGEBRAS
L*(G)
AND
A/(G).
BIBLIOGRAPHY
267
LIST
OF
SYMBOLS
271
INDEX
273 |
| any_adam_object | 1 |
| author | Pedersen, Gert Kjaergård 1940-2004 |
| author_GND | (DE-588)17230363X |
| author_facet | Pedersen, Gert Kjaergård 1940-2004 |
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| classification_tum | MAT 460f |
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| discipline | Mathematik |
| edition | Corr. 2. print. |
| format | Book |
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| id | DE-604.BV011146995 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-13T00:27:32Z |
| institution | BVB |
| isbn | 3540967885 0387967885 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007471629 |
| oclc_num | 34012089 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-188 DE-11 DE-703 |
| owner_facet | DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-188 DE-11 DE-703 |
| physical | XIV, 277 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Pedersen, Gert Kjaergård 1940-2004 Verfasser (DE-588)17230363X aut Analysis now Gert K. Pedersen Corr. 2. print. New York [u.a.] Springer 1995 XIV, 277 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 118 Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s DE-604 Graduate texts in mathematics 118 (DE-604)BV000000067 118 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007471629&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pedersen, Gert Kjaergård 1940-2004 Analysis now Graduate texts in mathematics Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 |
| title | Analysis now |
| title_auth | Analysis now |
| title_exact_search | Analysis now |
| title_full | Analysis now Gert K. Pedersen |
| title_fullStr | Analysis now Gert K. Pedersen |
| title_full_unstemmed | Analysis now Gert K. Pedersen |
| title_short | Analysis now |
| title_sort | analysis now |
| topic | Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functional analysis Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007471629&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT pedersengertkjaergard analysisnow |