Functional Analysis: Vol. I by Yurij M. BerezanskyFunctional Analysis: Vol. I by Yurij M. Berezansky

Functional Analysis: Vol. I

byYurij M. Berezansky, Zinovij G. Sheftel, Georgij F. Us

Paperback | October 10, 2011

Pricing and Purchase Info

$254.52 online 
$275.95 list price save 7%
Earn 1,273 plum® points

Prices and offers may vary in store

Quantity:

In stock online

Ships free on orders over $25

Not available in stores

about

"Functional Analysis" is a comprehensive, 2-volume treatment of a subject lying at the core of modern analysis and mathematical physics. The first volume reviews basic concepts such as the measure, the integral, Banach spaces, bounded operators and generalized functions. Volume II moves on to more advanced topics including unbounded operators, spectral decomposition, expansion in generalized eigenvectors, rigged spaces, and partial differential operators. This text provides students of mathematics and physics with a clear introduction into the above concepts, with the theory well illustrated by a wealth of examples. Researchers will appreciate it as a useful reference manual.
Title:Functional Analysis: Vol. IFormat:PaperbackDimensions:426 pagesPublished:October 10, 2011Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034899394

ISBN - 13:9783034899390

Reviews

Table of Contents

1 Measure Theory.- 1 Operations on Sets. Ordered Sets.- 1.1 Operations on Sets n2.- 1.2 Ordered Sets. The Zorn Lemma.- 2 Systems of Sets.- 2.1 Rings and Algebras of Sets.- 2.2 ?-Rings and ?-Algebras.- 2.3 Generated Rings and Algebras.- 3 Measure of a Set. Simple Properties of Measures.- 4 Outer Measure.- 5 Measurable Sets. Extension of a Measure.- 6 Properties of Measures and Measurable Sets.- 7 Monotone Classes of Sets. Uniqueness of Extensions of Measures.- 8 Measures Taking Infinite Values.- 9 Lebesgue Measure of Bounded Linear Sets.- 10 Lebesgue Measure on the Real Line.- 11 Lebesgue Measure in the N-Dimensional Euclidean Space.- 12 Discrete Measures.- 13 Some Properties of Nondecreasing Functions.- 13.1 Discontinuity Points of Monotone Functions.- 13.2 Jump Function. Continuous Part of a Nondecreasing Function.- 14 Construction of a Measure for a Given Nondecreasing Function. Lebesgue-Stieltjes Measure.- 15 Reconstruction of a Nondecreasing Function for a Given Lebesgue-Stieltjes Measure.- 16 Charges and Their Properties.- 16.1 Concept of a Charge. Decomposition in Hahn's Sense.- 16.2 Decomposition in Jordan's Sense.- 17 Relationship between Functions of Bounded Variation and Charges.- 2 Measurable Functions.- 1 Measurable Spaces. Measure Spaces. Measurable Functions.- 2 Properties of Measurable Functions.- 3 Equivalence of Functions.- 4 Sequences of Measurable Functions.- 5 Simple Functions. Approximation of Measurable Functions by Simple Functions. The Luzin Theorem.- 3 Theory of Integration.- 1 Integration of Simple Functions.- 2 Integration of Measurable Bounded Functions.- 3 Relationship Between the Concepts of Riemann and Lebesgue Integrals.- 4 Integration of Nonnegative Unbounded Functions.- 5 Integration of Unbounded Functions with Alternating Sign.- 6 Limit Transition under the Sign of the Lebesgue Integral.- 7 Integration over a Set of Infinite Measure.- 8 Summability and Improper Riemann Integrals.- 8.1 Integrals of Unbounded Functions.- 8.2 Integrals over Sets of Infinite Measure.- 9 Integration of Complex-Valued Functions.- 10 Integrals over Charges.- 10.1 Integrals over Charges.- 10.2 Integral over Complex-Valued Charges.- 11 Lebesgue-Stieltjes Integral and Its Relation to the Riemann-Stieltjes Integral.- 12 The Lebesgue Integral and the Theory of Series.- 4 Measures in the Products of Spaces. Fubini Theorem.- 1 Direct Product of Measurable Spaces. Sections of Sets and Functions.- 2 Product of Measures.- 3 The Fubini Theorem.- 4 Products of Finitely Many Measures.- 5 Absolute Continuity and Singularity of Measures, Charges, and Functions. Radon-Nikodym Theorem. Change of Variables in the Lebesgue Integral.- 1 Absolutely Continuous Measures and Charges.- 2 Radon-Nikodym Theorem.- 3 Radon-Nikodym Derivative. Change of Variables in the Lebesgue Integral.- 4 Mappings of Measure Spaces. Change of Variables in the Lebesgue Integral. (Another Approach).- 5 Singularity of Measures and Charges. Lebesgue Decomposition.- 6 Absolutely Continuous Functions. Basic Properties.- 7 Relationship Between Absolutely Continuous Functions and Charges.- 8 Newton-Leibniz Formula. Singular Functions. Lebesgue Decomposition of a Function of Bounded Variation.- 6 Linear Normed Spaces and Hilbert Spaces.- 1 Topological Spaces.- 2 Linear Topological Spaces.- 3 Linear Normed and Banach Spaces.- 4 Completion of Linear Normed Spaces.- 5 Pre-Hilbert and Hilbert Spaces.- 6 Quasiscalar Product and Seminorms.- 7 Examples of Banach and Hilbert Spaces.- 7.1 The Spaces ?N and ?N.- 7.2 The Space C(Q).- 7.3 The Space M(R).- 7.4 The Space Cm($$ \tilde G $$).- 7.5 The Space C?($$ \tilde G $$).- 8 Spaces of Summable Functions. Spaces Lp.- 8.1 Hölder and Minkowski Inequalities. Definition of the Spaces Lp.- 8.2 Everywhere Dense Sets in Lp. Separability Conditions.- 8.3 Different Types of Convergence in Lp.- 8.4 The Space lp.- 8.5 The Space L2(R,d?).- 8.6 Essentially Bounded Functions. The Space L?(R,d?).- 8.7 The Space l?.- 8.8 The Sobolev Spaces.- 7 Linear Continuous Functional and Dual Spaces.- 1 Theorem on an Almost Orthogonal Vector. Finite Dimensional Spaces.- 2 Linear Continuous Functional and Their Simple Properties. Dual Space.- 3 Extension of Linear Continuous Functionals.- 3.1 Extension by Continuity.- 3.2 Extension of a Functional Defined on a Subspace.- 4 Corollaries of the Hahn-Banach Theorem.- 5 General Form of Linear Continuous Functionals in Some Banach Spaces.- 5.1 The Concept of a Schauder Basis.- 5.2 The Space Dual to lp (1 <_20_3f_29_.-20_5.620_the20_spaces20_dual20_to20_l128_r2c_20_d3f_29_20_and20_l3f_28_r2c_20_d3f_29_.-20_5.720_the20_space20_dual20_to20_c28_q29_.-20_620_embedding20_of20_a20_linear20_normed20_space20_in20_the20_second20_dual20_space.20_reflexive20_spaces.-20_720_banach-steinhaus20_theorem.20_weak20_convergence.-20_7.120_banach-steinhaus20_theorem.-20_7.220_weak20_convergence20_of20_linear20_continuous20_functional.-20_7.3weak20_convergence20_in20_28_c28_5b_a2c_20_b5d_29_29_3f_.20_the20_helly20_theorems.-20_7.420_weak20_convergence20_in20_a20_linear20_normed20_space.-20_820_tikhonov20_product.20_weak20_topology20_in20_the20_dual20_space.-20_8.120_tikhonov20_product20_of20_topological20_spaces.-20_8.220_weak20_topology20_in20_the20_dual20_space.-20_920_orthogonality20_and20_orthogonal20_projections20_in20_hilbert20_spaces.20_general20_form20_of20_a20_linear20_continuous20_functional.-20_9.120_orthogonality.20_theorem20_on20_the20_projection20_of20_a20_vector20_onto20_a20_subspace.-20_9.220_orthogonal20_sums20_of20_subspaces.-20_9.320_linear20_continuous20_functionals20_in20_hilbert20_spaces.-20_1020_orthonormal20_systems20_of20_vectors20_and20_orthonormal20_bases20_in20_hilbert20_spaces.-20_10.120_orthonormal20_systems20_of20_vectors.20_the20_bessel20_inequality.-20_10.220_orthonormal20_bases20_in20_h.20_the20_parseval20_equality.-20_10.320_orthogonalization20_of20_a20_system20_of20_vectors.-20_10.420_examples20_of20_orthogonal20_polynomials.-20_10.520_orthonormal20_systems20_of20_vectors20_of20_arbitrary20_cardinality.-20_820_linear20_continuous20_operators.-20_120_linear20_operators20_in20_normed20_spaces.-20_220_the20_space20_of20_linear20_continuous20_operators.-20_320_product20_of20_operators.20_the20_inverse20_operator.-20_3.120_product20_of20_operators.-20_3.220_normed20_algebras.-20_3.320_the20_inverse20_operator.-20_420_the20_adjoint20_operator.-20_520_linear20_operators20_in20_hilbert20_spaces.-20_5.120_bilinear20_forms.-20_5.220_selfadjoint20_operators.-20_5.320_nonnegative20_operators.-20_5.420_projection20_operators.-20_5.520_normal20_operators.-20_5.620_unitary20_and20_isometric20_operators.-20_620_matrix20_representation20_of20_operators20_in20_hilbert20_spaces.-20_6.120_linear20_operators20_in20_a20_separable20_space.-20_6.220_selfadjoint20_operators.-20_6.320_nonnegative20_operators.-20_6.420_orthoprojectors.-20_6.520_isometric20_operators.-20_6.620_jacobian20_matrices.-20_720_hilbert-schmidt20_operators.-20_7.120_absolute20_norm.-20_7.220_integral20_hilbert-schmidt20_operators.-20_820_spectrum20_and20_resolvent20_of20_a20_linear20_continuous20_operator.-20_920_compact20_operators.20_equations20_with20_compact20_operators.-20_120_definition20_and20_properties20_of20_compact20_operators.-20_220_riesz-schauder20_theory20_of20_solvability20_of20_equations20_with20_compact20_operators.-20_320_solvability20_of20_fredholm20_integral20_equations.-20_3.120_some20_classes20_of20_integral20_operators.-20_3.220_solvability20_of20_fredholm20_integral20_equations20_of20_the20_second20_kind.-20_3.320_integral20_equations20_with20_degenerate20_kernels.-20_420_spectrum20_of20_a20_compact20_operator.-20_520_spectral20_radius20_of20_an20_operator.-20_5.120_power20_series20_with20_operator20_coefficients.-20_5.220_spectral20_radius20_of20_a20_linear20_continuous20_operator.-20_5.320_method20_of20_successive20_approximations.-20_620_solution20_of20_integral20_equations20_of20_the20_second20_kind20_by20_the20_method20_of20_successive20_approximations.-20_1020_spectral20_decomposition20_of20_compact20_selfadjoint20_operators.20_analytic20_functions20_of20_operators.-20_120_spectral20_decomposition20_of20_a20_compact20_selfadjoint20_operator.-20_1.120_one20_property20_of20_hermitian20_bilinear20_forms.-20_1.220_theorem20_on20_existence20_of20_an20_eigenvector20_for20_a20_selfadjoint20_compact20_operator.-20_1.320_spectral20_theorem20_for20_a20_compact20_selfadjoint20_operator.-20_220_integral20_operators20_with20_hermitian20_kernels.-20_2.120_spectral20_decomposition20_of20_a20_selfadjoint20_integral20_operator.-20_2.220_bilinear20_decomposition20_of20_hermitian20_kernels.-20_2.320_hilbert-schmidt20_theorem.-20_2.420_integral20_operators20_with20_positive20_definite20_kernels.20_the20_mercer20_theorem.-20_320_the20_bochner20_integral.-20_420_analytic20_functions20_of20_operators.-20_1120_elements20_of20_the20_theory20_of20_generalized20_functions.-20_120_test20_and20_generalized20_functions.-20_1.120_space20_of20_test20_functions20_d20_28_3f_n29_.-20_1.220_operators20_of20_averaging.-20_1.320_decomposition20_of20_the20_unit.-20_1.420_space20_of20_generalized20_functions20_d3f_28_3f_n29_.-20_1.520_order20_of20_a20_generalized20_function.-20_1.620_support20_of20_a20_generalized20_function.-20_1.720_regularization.-20_220_operations20_with20_generalized20_functions.-20_2.120_operations20_in20_d3f_28_3f_n29_.20_definitions.-20_2.220_multiplication20_of20_generalized20_functions20_by20_a20_smooth20_function.-20_2.320_change20_of20_variables20_in20_generalized20_functions.-20_2.420_differentiation20_of20_generalized20_functions.-20_320_tempered20_generalized20_functions.20_fourier20_transformation.-20_3.120_the20_space20_s28_3f_n29_20_of20_test20_28_rapidly20_decreasing29_20_functions.-20_3.220_the20_space20_s3f_20_28_3f_n29_20_of20_28_tempered29_20_generalized20_functions.-20_3.320_fourier20_transformation.-20_bibliographical20_notes.-20_references. .-="" 5.6="" the="" spaces="" dual="" to="" _l128_r2c_="" _d3f_29_="" and="" _l3f_28_r2c_="" _d3f_29_.-="" 5.7="" space="" _c28_q29_.-="" 6="" embedding="" of="" a="" linear="" normed="" in="" second="" space.="" reflexive="" spaces.-="" 7="" banach-steinhaus="" theorem.="" weak="" convergence.-="" 7.1="" theorem.-="" 7.2="" convergence="" continuous="" functional.-="" 7.3weak="" _28_c28_5b_a2c_="" _b5d_29_29_3f_.="" helly="" theorems.-="" 7.4="" space.-="" 8="" tikhonov="" product.="" topology="" 8.1="" product="" topological="" 8.2="" 9="" orthogonality="" orthogonal="" projections="" hilbert="" spaces.="" general="" form="" 9.1="" orthogonality.="" theorem="" on="" projection="" vector="" onto="" subspace.-="" 9.2="" sums="" subspaces.-="" 9.3="" functionals="" 10="" orthonormal="" systems="" vectors="" bases="" 10.1="" vectors.="" bessel="" inequality.-="" 10.2="" h.="" parseval="" equality.-="" 10.3="" orthogonalization="" system="" vectors.-="" 10.4="" examples="" polynomials.-="" 10.5="" arbitrary="" cardinality.-="" operators.-="" 1="" operators="" 2="" 3="" operators.="" inverse="" operator.-="" 3.1="" 3.2="" algebras.-="" 3.3="" 4="" adjoint="" 5="" 5.1="" bilinear="" forms.-="" 5.2="" selfadjoint="" 5.3="" nonnegative="" 5.4="" 5.5="" normal="" unitary="" isometric="" matrix="" representation="" 6.1="" separable="" 6.2="" 6.3="" 6.4="" orthoprojectors.-="" 6.5="" 6.6="" jacobian="" matrices.-="" hilbert-schmidt="" absolute="" norm.-="" integral="" spectrum="" resolvent="" compact="" equations="" with="" definition="" properties="" riesz-schauder="" theory="" solvability="" fredholm="" equations.-="" some="" classes="" kind.-="" degenerate="" kernels.-="" spectral="" radius="" an="" power="" series="" operator="" coefficients.-="" method="" successive="" approximations.-="" solution="" kind="" by="" decomposition="" analytic="" functions="" 1.1="" one="" property="" hermitian="" 1.2="" existence="" eigenvector="" for="" 1.3="" 2.1="" 2.2="" 2.3="" 2.4="" positive="" definite="" kernels.="" mercer="" bochner="" integral.-="" 11="" elements="" generalized="" functions.-="" test="" d="" _28_3f_n29_.-="" averaging.-="" unit.-="" 1.4="" _d3f_28_3f_n29_.-="" 1.5="" order="" function.-="" 1.6="" support="" 1.7="" regularization.-="" operations="" _d3f_28_3f_n29_.="" definitions.-="" multiplication="" smooth="" change="" variables="" differentiation="" tempered="" functions.="" fourier="" transformation.-="" _s28_3f_n29_="" _28_rapidly="" _decreasing29_="" _s3f_="" _28_3f_n29_="" _28_tempered29_="" bibliographical="" notes.-="">