Alphabeta MathBanach range invariance and EIT rates — Examples and Counterexamples
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1 · Prerequisites
- Banach range invariance and EIT rates
- Banach-Space Differential Calculus and Banach Manifolds
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Inverse maps, identifiability, and stability
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Nonlinear maps and tangential cone control
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simple Field Extensions and the Construction of the Complex Numbers
- Suprema and Infima
- The ZFC Axioms and the Basic Set Constructions
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Example
Range invariance can hold when the tangential cone condition fails
Example