Jerry's Digital Garden

Hi, welcome to my digital garden, where I collect math results that I learn.

Occasionally, I make up some definitions to play with. These definition blocks will always have a * at the end of its title to indicate that it lives mostly in my head. These terms will always be referenced with a link to their definition block.

Contents

Part 1: General Tools
- Chapter 1: Category Theory
  - circle1.1: Categories and Functors
  - circle1.2: Universal Construction
  - circle1.3: Universal Constructions for Modules
- Chapter 2: Functions
  - circle2.1: Gluing Lemmas
  - circle2.2: Preimages
- Chapter 3: Inequalities and Computations
  - circle3.1: Dyadic Rational Numbers
  - circle3.2: Product Inequalities
- circleChapter 4: Notations
Part 2: General Topology
- Chapter 5: Topological Spaces
  - circle5.1: Definitions
  - circle5.2: Filters
  - circle5.3: Nets
  - circle5.4: Neighbourhoods
  - circle5.5: Interior, Closure, Boundary
  - circle5.6: Continuous Maps
  - circle5.7: Product Spaces
  - circle5.8: Hausdorff Spaces
  - circle5.9: Regular Spaces
  - circle5.10: Normal Spaces
  - circle5.11: Quotient Topologies
  - circle5.12: Connectedness
  - circle5.13: Path-Connectedness
  - circle5.14: Local Path-Connectedness
  - circle5.15: Partitions of Unity
  - circle5.16: Compactness
  - circle5.17: $\sigma$-Compact Spaces
  - circle5.18: Paracompact Spaces
  - circle5.19: Support
  - 5.20: Locally Compact Hausdorff Spaces
    circle5.20.1: Paracompactness and LCH Spaces
  - circle5.21: Continuous Functions Vanishing at Infinity
  - circle5.22: Semicontinuity
  - circle5.23: Baire Spaces
  - circle5.24: Embeddings in Cubes
  - circle5.25: Compactifications
  - circle5.26: Open Preimage Functions
- Chapter 6: Uniform Spaces
  - circle6.1: Uniform Structures
  - circle6.2: Uniform Continuity
  - circle6.3: Pseudometrics
  - circle6.4: Compact Uniform Spaces
  - circle6.5: Equicontinuity
  - circle6.6: Cauchy Filters
  - circle6.7: Completeness
  - circle6.8: The Hausdorff Completion
- Chapter 7: Function Spaces
  - circle7.1: Ideals
  - circle7.2: Topologies With Respect to Ideals
  - circle7.3: The Uniform Topology
  - circle7.4: The Stone-Weierstrass Theorem
- Chapter 8: Metric Spaces
  - circle8.1: Metrics
  - circle8.2: Distance Between Sets
- Chapter 9: Topological Groups
  - circle9.1: Group Topologies
  - circle9.2: Coset Spaces and Factor Groups
- circleChapter 10: Notations
Part 3: Functional Analysis
- Chapter 11: Topological Vector Spaces
  - circle11.1: Vector Space Topologies
  - circle11.2: Complexification
  - circle11.3: Pseudonorms
  - circle11.4: Bounded Sets
  - circle11.5: The Dual Space
  - circle11.6: Continuous Linear Maps
  - circle11.7: Quotient Spaces
  - circle11.8: The Hausdorff Completion
  - circle11.9: Complete Metric TVSs
  - circle11.10: Projective Limits
  - circle11.11: Inductive Limits
  - circle11.12: Vector-Valued Function Spaces
  - circle11.13: Spaces of Linear Maps
  - circle11.14: Equicontinuous Families of Linear Maps
- Chapter 12: Locally Convex Spaces
  - circle12.1: Convex Sets and Seminorms
  - circle12.2: Continuous Linear Maps
  - circle12.3: Compact Convex Sets
  - circle12.4: Barreled Spaces
  - circle12.5: Bornological Spaces
  - circle12.6: Quotient Spaces
  - circle12.7: Projective Limits
  - 12.8: Inductive Limits
    circle12.8.2: Strict Inductive Limits
  - circle12.9: The Hahn-Banach Theorem
  - circle12.10: Locally Convex Spaces of Linear Maps
  - circle12.11: The Projective Tensor Product
- Chapter 13: Normed Vector Spaces
  - circle13.1: Norms
  - circle13.2: Conditional and Absolute Convergence
  - circle13.3: Linear Maps
  - circle13.4: Separable Normed Vector Spaces
  - circle13.5: Multilinear Maps
  - circle13.6: Hilbert Spaces
- Chapter 14: The Riemann-Stieltjes Integral
  - circle14.1: Partitions
  - circle14.2: Functions of Bounded Variation
  - circle14.3: Riemann-Stieltjes Sums and Integrals
  - circle14.4: Integrators of Bounded Variation
  - circle14.5: Path Integrals
  - circle14.6: Regulated Functions
  - circle14.7: The Lebesgue-Stieltjes Integral
- Chapter 15: $L^{p}$ Spaces
  - circle15.1: Basic Properties
  - circle15.2: $L^{p}$ Inequalities
  - circle15.3: Duality of $L^{p}$ Spaces
  - circle15.4: Uniform Integrability
  - circle15.5: $l^{p}$ Direct Sums
- Chapter 16: Order Structures
  - circle16.1: Vector Lattices
  - circle16.2: Banach Lattices
- Chapter 17: Duality
  - circle17.1: Dual Systems
  - circle17.2: Polars
  - circle17.3: The Mackey-Arens Theorem
- Chapter 18: Convex Functions
  - circle18.1: Convexity
  - circle18.2: Subgradients
  - circle18.3: Conjugate Functions
  - circle18.4: Support Functions
- Chapter 19: Interpolation Spaces
  - circle19.1: Compatible Couples
  - circle19.2: The Complex Interpolation Method
- circleChapter 20: Notations
Part 4: Measure Theory and Integration
- Chapter 21: Set Systems
  - circle21.1: Algebras
  - circle21.2: The Borel $\sigma$-Algebra
  - circle21.3: Lambda Systems
  - circle21.4: Elementary Families
  - circle21.5: Limits of Sets
- Chapter 22: Positive Measures
  - circle22.1: Measures
  - circle22.2: Complete Measures
  - circle22.3: Semifinite Measures
  - circle22.4: Scaffolds
  - circle22.5: $\sigma$-Finite Measures
  - circle22.6: Localisable Measures
  - circle22.7: Regular Measures
  - circle22.8: Carathéodory’s Extension Theorem
  - circle22.9: Lebesgue-Stieltjes Measures
  - circle22.10: Product Measures
  - circle22.11: Kolmogorov’s Extension Theorem
  - circle22.12: The Fréchet-Nikodym Metric
- Chapter 23: Signed, Complex, and Vector Measures
  - circle23.1: Signed and Complex Measures
  - circle23.2: Vector Measures
  - circle23.3: Total Variation
  - circle23.4: Absolute Continuity
  - circle23.5: Mutually Singular
  - circle23.6: The Lebesgue-Radon-Nikodym Theorem
  - circle23.7: Spaces of Finite Measures
- Chapter 24: Radon Measures
  - circle24.1: Radon Measures
  - circle24.2: The Riesz Representation Theorem
  - circle24.3: Dual of $C_{0}$
- Chapter 25: Measurable Functions
  - circle25.1: Measurable Functions
  - circle25.2: Product $\sigma$-Algebras
  - circle25.3: Real-Valued Measurable Functions
  - circle25.4: Simple Functions
  - circle25.5: Measurable Functions into Metric Spaces
  - circle25.6: Approximations with Simple Functions
  - circle25.7: Convergence in Measure
- Chapter 26: The Lebesgue Integral
- Chapter 27: The Bochner Integral
  - circle27.1: Strongly Measurable Functions
  - 27.2: The Bochner Integral
    circle27.2.1: Vector Measure Version
- Chapter 28: Integration on Locally Compact Groups
  - circle28.1: Locally Compact Groups
  - circle28.2: Haar Measures
  - circle28.3: The Modular Function
- circleChapter 29: Notations
Part 5: Calculus
- Chapter 30: Differential Calculus
  - circle30.1: Derivatives and Remainders
  - circle30.2: Differentiation With Respect to Sets
  - circle30.3: The Mean Value Theorem
  - circle30.4: Higher Derivatives
  - circle30.5: Taylor’s Formula
  - circle30.6: Partial Derivatives
  - circle30.7: Power Series
  - circle30.8: Inverse Mappings
  - circle30.9: Derivatives on $\mathbb{R}^{n}$
- Chapter 31: Complex Analysis
  - circle31.1: Complex Differentiability
  - circle31.2: Zeroes of Holomorphic Functions
  - circle31.3: The Riemann Sphere
  - circle31.4: Spaces of Holomorphic Functions
  - circle31.5: The Complex Logarithm
  - circle31.6: Entire Functions
  - circle31.7: Runge’s Theorem
- circleChapter 32: Notations
Part 6: Operator Algebras
- Chapter 33: Banach Algebras
  - circle33.1: Banach Algebras
  - circle33.2: Invertible Elements
  - circle33.3: The Index Group
  - circle33.4: Ideals
  - circle33.5: The Spectrum
  - circle33.6: The Holomorphic Functional Calculus
  - circle33.7: Multiplicative Functionals
  - circle33.8: The Gelfand Transform
- Chapter 34: $C^{*}$-Algebras
  - circle34.1: Involutions
  - circle34.2: Self-Adjoint Elements
  - circle34.3: Order Structures of $C^{*}$-Algebras
- Chapter 35: Examples of Operator Algebras
  - circle35.1: The Matrix Algebra
  - circle35.2: $B(H)$
  - circle35.3: $C_{0}(X)$
  - circle35.4: The Hardy Space
  - circle35.5: The Disk Algebra
  - circle35.6: $\ell^{1}(\integer)$
  - circle35.7: $BC(X)$
- circleChapter 36: Notations

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Recent Changes

circleTheorem 23.6.2: Lebesgue-Radon-Nikodym (Localisable)
circleTheorem 23.6.1: Lebesgue-Radon-Nikodym (Finite)
circleDefinition 22.12.1: Fréchet-Nikodym Metric
circleTheorem 22.11.3: Kolmogorov’s Extension Theorem
circleLemma 22.11.2: [Lemma 1.17, Bau14]
circleDefinition 22.11.1: Consistent
circleTheorem 22.10.3: Fubini-Tonelli Theorem
circleLemma 22.10.2: [Proposition 2.34, Fol99]
circleDefinition 22.10.1: Product Measure
circle23.6: The Lebesgue-Radon-Nikodym Theorem

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