Alphabeta MathBanach-Space Differential Calculus and Banach Manifolds
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Geometric Hahn Banach and Convex Separation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Approximation and Compactness in C(K)
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Fréchet derivative between Banach spaces
Lemma
The Fréchet derivative is unique
Theorem
Chain sum product and composition rules for Banach derivatives
Definition
C k map between Banach spaces
Lemma
Banach mean value estimate on a convex set
Theorem
Inverse function theorem for Banach spaces
Theorem
Implicit function theorem for Banach spaces
Definition
Countable base Banach manifold and smooth map
Definition
Tangent space and differential on a Banach manifold
Lemma
Banach manifold differentials are chart independent
Definition
Split Banach submanifold
Theorem
Regular value theorem for Banach manifolds
Definition
Smooth Banach vector bundle and section
Theorem
A transverse Banach bundle section has a split zero submanifold
Definition
Fredholm map between Banach manifolds
Lemma
Local finite-dimensional reduction for a Fredholm map
Proposition
The index of a Fredholm map is locally constant
Remark
Surjectivity alone does not imply a complemented kernel
Sources
- Zuoqin Wang, Lecture 6: Differential Calculus on Banach Spaces — §2.1
- Zuoqin Wang, Lecture 6 — §2.1 (Exercise 6)
- Zuoqin Wang, Lecture 6 — §§2.1.1–2.1.4
- Zuoqin Wang, Lecture 6 — §2.3
- Zuoqin Wang, Lecture 6 — §2.2.3 (mean value theorem via Hahn–Banach)
- Zuoqin Wang, Lecture 6 — §§3.1–3.2 (contraction proof, with derivative normalization)
- Zuoqin Wang, Lecture 6 — §3.3 (implicit theorem via the map (x,y) ↦ (x,F(x,y)))
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §1.3 and §2.11
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §1.3
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.11 (regular values as onto with complemented kernel)
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.11, regular values
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.12 (vector bundles and sections)
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.12
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.11 (Fredholm maps)
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.11, Theorem 2.19 proof
- Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex — §2.11