Functional Analysis for Physics and Engineering

Content: Prologue What Functional Analysis tells usFrom perspective of the limit From perspective of infinite dimension From perspective of quantum mechanical theoryTopology Fundamentals Continuous mapping Homeomorphism Vector spaceWhat is vector space?Property of vector spaceHierarchy of vector spaceHilbert spaceBasis and completenessEquivalence of L2 spaces with â 2 spacesTensor spaceTwo faces of one tensor"Vector" as a linear functionTensor as a multilinear functionComponent of tensorLebesgue integralMotivation & MeritsMeasure theoryLebesgue integralLebesgue convergence theoremLp spaceWavelet 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 sequenceA.1 Convergence of real sequenceA.2 Bounded sequenceA.3 Uniqueness of the limit of real sequence Cauchy sequenceB.1 What is Cauchy sequence?B.2 Cauchy criterion for real number sequence Real number seriesC.1 Limit of real number seriesC.2 Cauchy criterion for real number seriesContinuity and smoothness of function D.1 Limit of function D.2 Continuity of function D.3 Derivative of function D.4 Smooth functionFunction sequenceE.1 Pointwise convergenceE.2 Uniform convergenceE.3 Cauchy criterion for function seriesF Uniformly convergent sequence of functionsF.1 Continuity of the limit functionF.2 Integrability of the limit functionF.3 Differentiability of the limit functionG Function seriesG.1 Infinite series of functionsG.2 Properties of uniformly convergent series of functionsH Matrix eigenvalue problemH.1 Eigenvalue and eigenvectorH.2 Hermite matrixI Eigenspace decompositionI.1 Eigenspace of matrixI.2 Direct sum decompositionIndex