E-Book, Englisch, 285 Seiten
Shima Functional Analysis for Physics and Engineering
Erscheinungsjahr 2015
ISBN: 978-1-4822-2303-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
An Introduction
E-Book, Englisch, 285 Seiten
ISBN: 978-1-4822-2303-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with graduate-level mathematics. Topics include topology, vector spaces, tensor spaces, Lebesgueintegrals, and operators, to name a few. Two central issues—the theory of Hilbert space and the operator theory—and how they relate to quantum physics are covered extensively. Each chapter explains, concisely, the purpose of the specific topic and the benefit of understanding it. Researchers and graduate students in physics, mechanical engineering, and information science will benefit from this view of functional analysis.
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Prologue
What Functional Analysis tells us
From perspective of the limit
From perspective of infinite dimension
From perspective of quantum mechanical theory
Topology
Fundamentals
Continuous mapping
Homeomorphism
Vector space
What is vector space?
Property of vector space
Hierarchy of vector space
Hilbert space
Basis and completeness
Equivalence of L2 spaces with l2 spaces
Tensor space
Two faces of one tensor
"Vector" as a linear function
Tensor as a multilinear function
Component of tensor
Lebesgue integral
Motivation & Merits
Measure theory
Lebesgue integral
Lebesgue convergence theorem
Lp space
Wavelet
Continuous wavelet analysis
Discrete wavelet analysis
Wavelet space
Distribution
Motivation & Merits
Establishing the concept of distribution
Examples of distribution
Mathematical manipulation of distribution
Completion
Completion of number space
Sobolev space
Operator
Classification of operators
Essence of operator theory
Preparation toward eigenvalue-like problem
Practical importance of non-continuous operators
Real number sequence
A.1 Convergence of real sequence
A.2 Bounded sequence
A.3 Uniqueness of the limit of real sequence
Cauchy sequence
B.1 What is Cauchy sequence?
B.2 Cauchy criterion for real number sequence
Real number series
C.1 Limit of real number series
C.2 Cauchy criterion for real number series
Continuity and smoothness of function
D.1 Limit of function
D.2 Continuity of function
D.3 Derivative of function
D.4 Smooth function
Function sequence
E.1 Pointwise convergence
E.2 Uniform convergence
E.3 Cauchy criterion for function series
F Uniformly convergent sequence of functions
F.1 Continuity of the limit function
F.2 Integrability of the limit function
F.3 Differentiability of the limit function
G Function series
G.1 Infinite series of functions
G.2 Properties of uniformly convergent series of functions
H Matrix eigenvalue problem
H.1 Eigenvalue and eigenvector
H.2 Hermite matrix
I Eigenspace decomposition
I.1 Eigenspace of matrix
I.2 Direct sum decomposition
Index