Mayorga-Zambrano J. A course of functional analysis with calculus of variations

Juan Ricardo Mayorga Zambrano, 2023. — 382 p.Preface
Functional Analysis
Preliminaries
Sets
Relations and functions
Families of sets, partitions and equivalence relations
Order relations
Cardinality
Linear spaces and algebras
Linear operators
Problems
An introduction to topological spaces
Definition of topology
Topological basis and fundamental systems
Interior, adherence and boundary of a set. Density.
Separation conditions and sequences
Accumulation points. Limit superior and inferior.
Subspaces and continuity
Initial topology. Product spaces.
Compact sets
Problems
An introduction to metric spaces
Definition of metric
Definitions of norm and interior product
Basic properties of metric spaces
Complete metric spaces. Baire's theorem.
Isometries. Completion of a metric space.
The spaces [[a,b]]1 and [[a,b]]2
Banach fixed point theorem
Compact sets in metric spaces
Problems
Banach and Hilbert spaces
Introduction
Equivalence of norms
Finite-dimensional normed spaces. Weierstrass Approximation Theorem.
Additional examples of Banach and Hilbert spaces
The space ([n],p)
The space lp(R)
The Lebesgue space [I]p, IR
The space [[a,b]]n
The Sobolev spaces [I]1 and [I]1p
Schauder and Hilbert basis
Direct sums and orthogonal of a set
Hilbert basis of [I]2
Fourier-Legendre series
Trigonometric Fourier series. Stone-Weierstrass theorem.
Fourier-Hermite series
Convex sets, functions and hyperplanes
Problems
Fundamental theorems of Functional Analysis
Introduction
Some properties of L(V,W)
Continuous embeddings
Riesz-Fréchet representation theorem
Hahn-Banach theorem
The dual of [[a,b]] . The Riemann-Stieljes integral.
Geometric forms of Hahn-Banach theorem
Adjoint operator
Reflexivity and separability (I)
Uniform boundedness principle
The open mapping theorem and the closed graph theorem
Problems
Weak topologies, reflexivity and separability
Weak convergence and topology (E,E')
Weak * convergence and topology (E',E)
Reflexivity and separability (II)
Uniformly convex spaces
Problems
An introduction to the Calculus of Variations
Calculus on normed spaces
What Calculus of Variations is about
A problem from classical Calculus
A situation to deal with Calculus of Variations
Euler's finite-difference method
A couple of important things
Normed and Banach algebras
Exponential mapping
Small o
Uniform continuity. Canonical basis of [n]
Problems
The differential
Directional derivatives and Gateaux differential
Fréchet differential
Gateaux or not-Gateaux? That's the question
Differentiability of a generalized vector field
Examples
Problems
Chain rule. Class C1.
Chain rule
Mappings of class C1
Problems
Critical points and extremum
Definitions
Necessary condition for an extremum
Problems
Mean Value Theorem. Connectedness.
Classical and generalized Mean Value Theorem
Derivative of a curve
Main tool
Proof of the Mean Value Theorem
An important result that depends on connectedness
Problems
Some useful lemmas
Problems
Partial differentials
Differential in terms of partial differentials
Partial differentials and the class C1
Problems
Riemann integral
Step and regulated mappings
Integral of step and regulated mappings
Fundamental Theorem of Calculus and derivation under the integral sign
Problems
Taylor expansion. Second order conditions for extremum.
Higher-order differentials
Multilinear mappings
Higher-order differentials and multilinear mappings
Taylor expansions
Second-order conditions for extrema
Examples
Problems
The Euler-Lagrange equation and other classical topics
The elementary problem of the Calculus of Variations
The functional of the elementary problem. Admissible elements.
Weak and strong extremum
Derivation of Euler-Lagrange equation
Bernstein's existence and unicity theorem
Differentiability of solutions of the Euler-Lagrange equation
Convexity and minimizers
Second form of the Euler-Lagrange equation
Special cases
Examples
Problems
Some generalizations of the elementary problem of the Calculus of Variations.
The fixed end-point problem for n unknown functions
The isoperimetric problem
A generalization of the isoperimetric problem
Finite subsidiary conditions
Problems
References
Index