Linear and nonlinear functional analysis with applications: with 401 problems and 52 figures
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| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
SIAM
2013
|
| Series: | Other titles in applied mathematics
130 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIV, 832 S. graph. Darst. |
| ISBN: | 9781611972580 |
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| 100 | 1 | |a Ciarlet, Philippe G. |d 1938- |e Verfasser |0 (DE-588)143368362 |4 aut | |
| 245 | 1 | 0 | |a Linear and nonlinear functional analysis with applications |b with 401 problems and 52 figures |c Philippe G. Ciarlet |
| 264 | 1 | |a Philadelphia |b SIAM |c 2013 | |
| 300 | |a XIV, 832 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Other titles in applied mathematics |v 130 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Functional analysis |v Textbooks | |
| 650 | 4 | |a Nonlinear functional analysis |v Textbooks | |
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| 689 | 0 | 0 | |a Funktionalanalysis |0 (DE-588)4018916-8 |D s |
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| 830 | 0 | |a Other titles in applied mathematics |v 130 |w (DE-604)BV023088396 |9 130 | |
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| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026802275&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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| _version_ | 1804151433829810176 |
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| adam_text | This single-volume textbook covers the fundamentals of linear and nonlinear
functional analysis, illustrating most of the basic theorems with numerous
applications to linear and nonlinear partial differential equations and to selected
topics from numerical analysis and optimization theory.
This book has pedagogical appeal because it features
*
self-4»ntainedarKlœmpleteprcofsofmostofthetheoreni8,8omeo
are not always easy to locate in the literature or are difficult to reconstitute;
* 401
problems and
52
figures;
*
historical notes and original references that provide an idea of die genesis of
the important results; and
*
most of the core topics from linear and nonlinear functional analysis.
It is intended for advanced undergraduates, graduate students, and researchers and is
ideal for teaching or self-study.
Philippe G. Ciarlet began his academic career at the
Université
Pierre et
Marie
Curie, Paris, in
1974,
and moved to City University of Hong Kong in
2002.
He is
a member of eight academies, including the French Academy of Sciences and the
Chinese Academy of Sciences and of the Hong Kong Institution of
Science, and he is a Fellow of SIAM and the
AMS.
P. G. Ciarlet is
the recipient of a Grand Prize from the French Academy of Sciences
and
a
Humboldt
Research Award, as well as many other awards. He
is Doctor Honoris Causa, or Honorary Professor, at eight universities
and the author of
190
research papers and
15
boob.
CONTENTS
Preface
хш
1
Real Analysis and Theory of Functions: A Quick Review
1
Introduction
........................................ 1
1.1
Sets
......................................... 2
1.2
Mappings
...................................... 3
1.3
The axiom of choice and Zorn s lemma
...................... 5
1.4
Construction of the sets
R
and
С
......................... 8
1.5
Cardinal numbers; finite and infinite sets
..................... 9
1.6
Topological spaces
................................. 11
1.7
Continuity in topological spaces
.......................... 14
1.8
Compactness in topological spaces
........................ 15
1.9
Connectedness and simple-connectedness in topological spaces
......... 16
1.10
Metric spaces
.................................... 18
1.11
Continuity and uniform continuity in metric spaces
............... 21
1.12
Complete metric spaces
.............................. 22
1.13
Compactness in metric spaces
........................... 23
1.14
The Lebesgue measure in Rn; measurable functions
............... 25
1.15
The Lebesgue integral in Rn; the basic theorems
................ 28
1.16
Change of variable in Lebesgue integrals in Rn
................. 33
1.17
Volumes, areas, and lengths in Rn
....................... . 34
1.18
The spaces
Ст{П)
and
С™
(Ω);
domains in Mn
.................. 36
2
Normed Vector Spaces
43
Introduction
........................................ 43
2.1
Vector spaces;
Hamel
bases; dimension of a vector space
............ 44
2.2
Normed vector spaces; first properties and examples; quotient spaces
..... 47
2.3
The space
C(K;
Y)
with
К
compact; uniform convergence and local uniform
convergence
..................................... 53
2.4
The spaces £p,
1 <
p
<
oo
............................. 57
2.5
The Lebesgue spaces
¿^(Ω),
1 <
ρ
<
oo
......................
61
2.6
Regularization and approximation in the spaces
LP{ÇÏ),
1 <
ρ
<
oo
......
68
2.7
Compactness and finite-dimensional normed vector spaces; F. Riesz theorem
. 76
2.8
Application of compactness in finite-dimensional normed vector spaces: The
fundamental theorem of algebra
.......................... 79
vii
viii Contents
2.9
Continuous linear operators in normed vector spaces; the spaces C(X K),
£(X), and
Λ
.................................... 82
2.10
Compact linear operators in normed vector spaces
............... 89
2.11
Continuous multilinear mappings in normed vector spaces; the space
Ck{X1,X2,...,Xk;Y)
............................... 91
2.12
Korovkin s theorem
................................. 97
2.13
Application of Korovkin s theorem to polynomial approximation; Bohman s,
Bernstein s, and
Weierstraß
theorems
...................... 100
2.14
Application of Korovkin s theorem to trigonometric polynomial
approximation;
Fejér s
theorem
.......................... 104
2.15
The Stone-
Weierstraß
theorem
.......................... 109
2.16
Convex sets
..................................... 114
2.17
Convex functions
.................................. 118
3
Banach Spaces
123
Introduction
........................................ 123
3.1
Banach spaces; first properties
.......................... 124
3.2
First examples of Banach spaces; the spaces C(K; Y) with
К
compact and
Y
complete, and C(X;Y) with
Y
complete
..................... 130
3.3
Integral of a continuous function of a real variable with values in a Banach
space
......................................... 133
3.4
Further examples of Banach spaces: the spaces fP and LP(Q),
1 <
ρ
<
oo
. . . 135
3.5
Dual of a normed vector space; first examples; F. Riesz representation theorem
in
£Ρ(Ω),
1 <
ρ
<
с»
................................ 138
3.6
Series in Banach spaces
.............................. 148
3.7
Banach fixed point theorem
............................ 152
38
Application of Banach fixed point theorem: Existence of solutions to
nonlinear ordinary differential equations; Cauchy-Lipschitz theorem;
the pendulum equation
............................... 156
3.9
Application of Banach fixed point theorem: Existence of solutions to nonlinear
two-point boundary value problems
........................ 161
3.10
Ascoli-Arzelà s
theorem
.............................. 164
3.11
Application of
Ascoli-Arzelà s
theorem: Existence of solutions to nonlinear
ordinary differential equations; Cauchy-Peano theorem; Euler s method
. . . 169
4
Inner-Product Spaces and Hilbert Spaces
173
Introduction
........................................ 173
4.1
Inner-product spaces and Hilbert spaces; first properties;
Cauchy-Schwarz-Bunyakovskiï
inequality; parallelogram law
......... 174
4.2
First examples of inner-product spaces and Hilbert spaces; the spaces £2
and L2(Q)
...................................... 181
4.3
The projection theorem
.............................. 183
4.4
Application of the projection theorem: Least-squares solution of a linear
system
........................................ 193
4.5
Orthogonality: direct sum theorem
........................ 195
Contents ix
4.6 F. Riesz
representation theorem in a Hubert space
............... 197
4.7
First applications of the F. Riesz representation theorem: Hahn-Banach
theorem in a Hilbert space; adjoint operators; reproducing kernels
...... 199
4.8
Maximal
orthonormal
families in an inner-product space
............ 205
4.9
Hilbert bases and Fourier series in a Hilbert space
............... 213
4.10
Eigenvalues and eigenvectors of self-adjoint operators in inner-product spaces
219
4.11
The spectral theorem for compact self-adjoint operators
............ 221
5
The Great Theorems of Linear Functional Analysis
231
Introduction
........................................ 231
5.1
Baire s theorem; a first application: Noncompleteness of the space of all
polynomials
..................................... 232
5.2
Application of Baire s theorem: Existence of nowhere differentiable continuous
functions
...................................... 236
5.3
Banach-Steinhaus theorem, alias the uniform boundedness principle;
application to numerical quadrature formulas
.................. 238
5.4
Application of the Banach-Steinhaus theorem: Divergence of
Lagrange
interpolation
.................................... 245
5.5
Application of the Banach-Steinhaus theorem: Divergence of Fourier series
. 252
5.6
Banach open mapping theorem; a first application: Well-posedness of two-
point boundary value problems
.......................... 255
5.7
Banach closed graph theorem; a first application: Hellinger-Toeplitz theorem
259
5.8
The Hahn-Banach theorem in a vector space
.................. 261
5.9
The Hahn-Banach theorem in a normed vector space; first consequences
. . . 264
5.10
Geometric forms of the Hahn-Banach theorem; separation of convex sets
. . . 272
5.11
Dual operators; Banach closed range theorem
.................. 277
5.12
Weak convergence and weak
*
convergence
.................... 286
5.13
Banach-Saks-Mazur theorem
........................... 294
5.14
Reflexive spaces; the
Banach-Eberlein-Šmulian
theorem
............ 297
6
Linear Partial Differential Equations
305
Introduction
........................................ 305
6.1
Quadratic minimization problems; variational equations and variational
inequalities
..................................... 306
6.2
The Lax-Milgram lemma
............................. 310
6.3
Weak partial derivatives in
¿¿^(Ω);
a brief incursion into distribution theory
312
6.4
Hypoellipticity of
Δ
................................ 319
6.5
The Sobolev spaces Wm p{Q) and Hm
(Ω):
First properties
........... 326
6.6
The Sobolev spaces Wm P{Q) and
Ητη(Ω)
with
Ω
a domain; imbedding
theorems, traces, Green s formulas
........................ 331
6.7
Examples of second-order linear elliptic boundary value problems; the
membrane problem
................................. 338
6.8
Examples of fourth-order linear boundary value problems; the biharmonic
and plate problems
................................. 355
χ
Contents
6.9
Examples of nonlinear boundary value problems associated with variational
inequalities; obstacle problems
.......................... 363
6.10
Eigenvalue problems for second-order elliptic operators
............. 369
6.11
The spaces W^ii) and
Я~т(П);
J.L. Lions lemma
............. 377
6.12
The
Babuška-Brezzi inf-sup
theorem; application to constrained quadratic
minimization problems
............................... 382
6.13
Application of the
Babuška-Brezzi inf-sup
theorem: Primal, mixed, and dual
formulations of variational problems
....................... 388
6.14
Application of the
Babuška-Brezzi
inf-sup theorem and of J.L. Lions lemma:
The Stokes equations
................................ 394
6.15
A second application of J.L. Lions lemma:
Korn
s
inequality
.......... 403
6.16
Application of Korn s inequality: The equations of three-dimensional linearized
elasticity
....................................... 412
6.17
The classical
Poincaré
lemma and its weak version as an application of
J.L. Lions lemma and of the hypoellipticity of
Δ
................ 419
6.18
Application of
Poíncaré s
lemma: The classical and weak Saint-
Venant
lemmas;
the
Cesàro-Volterra
path integral formula
.................... 429
6.19
Another application of J.L. Lions lemma: The
Donati
lemmas
......... 437
6.20
Pfaff systems
.................................... 444
7
Differential Calculus in Normed Vector Spaces
451
Introduction
........................................ 451
7.1
The
Fréchet
derivative; the chain rule; the
Piola
identity; application to
extrema
of real-valued functions
.......................... 452
7.2
The mean value theorem in a normed vector space; first applications
..... 465
7.3
Application of the mean value theorem: Differentiability of the limit of a
sequence of differentiable functions
........................ 469
7.4
Application of the mean value theorem: Differentiability of a function defined
by an integral
.................................... 472
7.5
Application of the mean value theorem: Sard s theorem
............ 474
7.6
A mean value theorem for functions of class C1 with values in a Banach space
477
7.7
Newton s method for solving nonlinear equations; the Newton-Kantorovich
theorem in a Banach space
............................ 478
7.8
Higher order derivatives;
Schwarz
lemma.....................
500
7.9
Taylor formulas; application to
extrema
of real-valued functions
........ 507
7.10
Application: Maximum principle for second-order linear elliptic operators
. . 513
7.11
Application:
Lagrange
interpolation in Rn and multipoint Taylor formulas
. . 522
7.12
Convex functions and differentiability; application to
extrema
of real-valued
functions
...................................... 540
7.13
The implicit function theorem; first application: Class C°° of the mapping
A-+A-1
...................................... 548
7.14
The local inversion theorem; the
invariance
of domain theorem for mappings
of class Cl in Banach spaces; class C°° of the mapping A
—»
A1/2
....... 554
7.15
Constrained
extrema
of real-valued functions;
Lagrange
multipliers
...... 560
7.16
Lagrangians and saddle-points; primal and dual problems
........... 565
Contents xi
8 Differential
Geometry
in Rn 575
Introduction
........................................ 575
8.1
Curvilinear coordinates in an open subset of Rn
................. 576
8.2
Metric tensor; volumes and lengths in curvilinear coordinates
......... 578
8.3
Covariant derivative of a vector field
....................... 583
8.4
Tensors
—
a brief introduction
........................... 588
8.5
Necessary conditions satisfied by the metric tensor; the Riemann curvature
tensor
........................................ 595
8.6
Existence of an immersion on an open subset of Rn with a prescribed metric
tensor; the fundamental theorem of Riemannian geometry
........... 598
8.7
Uniqueness up to isometries of immersions with the same metric tensor;
the rigidity theorem for an open subset of
Шп
.................. 608
8.8
Curvilinear coordinates on a surface in K3
.................... 613
8.9
First fundamental form of a surface; areas, lengths, and angles on a surface
. 614
8.10
Isometric, equiareal, and
conformai
surfaces
................... 622
8.11
Second fundamental form of a surface; curvature on a surface
......... 624
8.12
Principal curvatures; Gaussian curvature
..................... 629
8.13
Covariant derivatives of a vector field defined on a surface; the
Gauß
and
Weingarten
formulas
................................ 636
8.14
Necessary conditions satisfied by the first and second fundamental forms: The
Gauß
and Codazzi-Mainardi equations
..................... 640
8.15 Gauß
Theorema
Egregium; application to cartography
............. 643
8.16
Existence of a surface with prescribed first and second fundamental forms;
the fundamental theorem of surface theory
.................... 646
8.17
Uniqueness of surfaces with the same fundamental forms; the rigidity theorem
for surfaces
..................................... 654
9
The Great Theorems of Nonlinear Functional Analysis
657
Introduction
........................................ 657
9.1
Nonlinear partial differential equations as the Euler-Lagrange equations
associated with the minimization of a functional
................ 658
9.2
Convex functions and sequentially lower semicontinuous functions with values
in
R U {oo}
..................................... 664
9.3
Existence of minimizers for coercive and sequentially weakly lower
semicontinuous functionals
............................. 671
9.4
Application to the
von Kármán
equations
.................... 674
9.5
Existence of minimizers in W^p(u)
........................ 683
9.6
Application to the p-Laplace operator
...................... 691
9.7
Polyconvexity; compensated compactness; John Ball s existence theorem in
nonlinear elasticity
................................. 693
9.8
Ekeland s variational principle; existence of minimizers for functionals that
satisfy the Palais-Smale condition
........................ 711
9.9
Brouwers
fixed point theorem
—
a first proof
.................. 718
9.10
Application of Brouwer s theorem to the
von Kármán
equations, by means of
the Galerkin method
................................ 726
xii Contents
9.11 Application
of Brouwer s theorem to the Navier-Stokes equations, by means
of the Galerkin method
.............................. 728
9.12
Schauder s fixed point theorem;
Schäfer s
fixed point theorem; Leray-Schauder
fixed point theorem
................................. 734
9.13
Monotone operators
................................ 739
9.14
The Minty-Browder theorem for monotone operators; application to the
p-Laplace operator
................................. 742
9.15
The
Brouwer topological
degree in M : Definition and properties
....... 748
9.16
Brouwer s fixed point theorem
—
a second proof
—
and the hairy ball theorem
764
9.17
Borsuk s and Borsuk-Ulam theorems; Brouwer s
invariance
of domain
theorem
....................................... 767
Bibliographical Notes
777
Bibliography
781
Main Notations
807
Index
815
|
| any_adam_object | 1 |
| author | Ciarlet, Philippe G. 1938- |
| author_GND | (DE-588)143368362 |
| author_facet | Ciarlet, Philippe G. 1938- |
| author_role | aut |
| author_sort | Ciarlet, Philippe G. 1938- |
| author_variant | p g c pg pgc |
| building | Verbundindex |
| bvnumber | BV041353768 |
| callnumber-first | Q - Science |
| callnumber-label | QA320 |
| callnumber-raw | QA320 |
| callnumber-search | QA320 |
| callnumber-sort | QA 3320 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 600 |
| classification_tum | MAT 460f |
| ctrlnum | (OCoLC)864910742 (DE-599)BVBBV041353768 |
| dewey-full | 515/.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.7 |
| dewey-search | 515/.7 |
| dewey-sort | 3515 17 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV041353768 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:54:45Z |
| institution | BVB |
| isbn | 9781611972580 |
| language | English |
| lccn | 2013018736 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026802275 |
| oclc_num | 864910742 |
| open_access_boolean | |
| owner | DE-20 DE-355 DE-BY-UBR DE-706 DE-188 DE-703 DE-83 DE-91G DE-BY-TUM DE-384 |
| owner_facet | DE-20 DE-355 DE-BY-UBR DE-706 DE-188 DE-703 DE-83 DE-91G DE-BY-TUM DE-384 |
| physical | XIV, 832 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | SIAM |
| record_format | marc |
| series | Other titles in applied mathematics |
| series2 | Other titles in applied mathematics |
| spelling | Ciarlet, Philippe G. 1938- Verfasser (DE-588)143368362 aut Linear and nonlinear functional analysis with applications with 401 problems and 52 figures Philippe G. Ciarlet Philadelphia SIAM 2013 XIV, 832 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Other titles in applied mathematics 130 Includes bibliographical references and index Functional analysis Textbooks Nonlinear functional analysis Textbooks Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionalanalysis (DE-588)4018916-8 s DE-604 Other titles in applied mathematics 130 (DE-604)BV023088396 130 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026802275&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026802275&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Ciarlet, Philippe G. 1938- Linear and nonlinear functional analysis with applications with 401 problems and 52 figures Other titles in applied mathematics Functional analysis Textbooks Nonlinear functional analysis Textbooks Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4123623-3 |
| title | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures |
| title_auth | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures |
| title_exact_search | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures |
| title_full | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures Philippe G. Ciarlet |
| title_fullStr | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures Philippe G. Ciarlet |
| title_full_unstemmed | Linear and nonlinear functional analysis with applications with 401 problems and 52 figures Philippe G. Ciarlet |
| title_short | Linear and nonlinear functional analysis with applications |
| title_sort | linear and nonlinear functional analysis with applications with 401 problems and 52 figures |
| title_sub | with 401 problems and 52 figures |
| topic | Functional analysis Textbooks Nonlinear functional analysis Textbooks Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functional analysis Textbooks Nonlinear functional analysis Textbooks Funktionalanalysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026802275&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026802275&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023088396 |
| work_keys_str_mv | AT ciarletphilippeg linearandnonlinearfunctionalanalysiswithapplicationswith401problemsand52figures |