Jerry's Digital Garden
Hi, welcome to my digital garden, where I collect math results that I learn.
Despite the contents being presented in a linear order by the table of contents, I will frequently reference things between chapters and sections.
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
- 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
- circle5.21: 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
- circle5.27: Extremely Disconnected Spaces
- 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
- Chapter 8: Metric Spaces
- circle8.1: Metrics
- circle8.2: Distance Between Sets
- Chapter 9: Topological Groups
- Chapter 10: Polish Spaces and Analytic Sets
- circle10.1: Polish Spaces
- circle10.2: Analytic Sets
- circle10.3: Zero Dimensional Spaces
- circleChapter 11: Notations
- Chapter 5: Topological Spaces
- Part 3: Functional Analysis
- Chapter 12: Topological Vector Spaces
- circle12.1: Vector Space Topologies
- circle12.2: Complexification
- circle12.3: Pseudonorms
- circle12.4: Bounded Sets
- circle12.5: The Dual Space
- circle12.6: Continuous Linear Maps
- circle12.7: Quotient Spaces
- circle12.8: The Hausdorff Completion
- circle12.9: Complete Metric TVSs
- circle12.10: Projective Limits
- circle12.11: Inductive Limits
- circle12.12: Vector-Valued Function Spaces
- circle12.13: Spaces of Linear Maps
- circle12.14: Equicontinuous Families of Linear Maps
- Chapter 13: Locally Convex Spaces
- circle13.1: Convex Sets and Seminorms
- circle13.2: Continuous Linear Maps
- circle13.3: Compact Convex Sets
- circle13.4: Barreled Spaces
- circle13.5: Bornological Spaces
- circle13.6: Quotient Spaces
- circle13.7: Projective Limits
- 13.8: Inductive Limits
- circle13.9: The Hahn-Banach Theorem
- circle13.10: Locally Convex Spaces of Linear Maps
- circle13.11: The Projective Tensor Product
- circle13.12: Nuclear Operators
- circle13.13: Nuclear Spaces
- Chapter 14: Normed Vector Spaces
- Chapter 15: The Riemann-Stieltjes Integral
- Chapter 16: $L^{p}$ Spaces
- Chapter 17: Order Structures
- circle17.1: Ordered Vector Spaces
- circle17.2: The Order Dual
- circle17.3: Vector Lattices
- circle17.4: Banach Lattices
- Chapter 18: Duality
- circle18.1: Dual Systems
- circle18.2: Polars
- circle18.3: The Mackey-Arens Theorem
- Chapter 19: Convex Functions
- circle19.1: Convexity
- circle19.2: Subgradients
- circle19.3: Conjugate Functions
- circle19.4: Support Functions
- Chapter 20: Interpolation Spaces
- circleChapter 21: Notations
- Chapter 12: Topological Vector Spaces
- Part 4: Measure Theory and Integration
- Chapter 22: Set Systems
- circle22.1: Algebras
- circle22.2: The Borel $\sigma$-Algebra
- circle22.3: Lambda Systems
- circle22.4: Elementary Families
- circle22.5: Limits of Sets
- Chapter 23: Positive Measures
- circle23.1: Measures
- circle23.2: Complete Measures
- circle23.3: Semifinite Measures
- circle23.4: Scaffolds
- circle23.5: $\sigma$-Finite Measures
- circle23.6: Localisable Measures
- circle23.7: Regular Measures
- circle23.8: Carathéodory’s Extension Theorem
- circle23.9: Lebesgue-Stieltjes Measures
- circle23.10: Product Measures
- circle23.11: Kolmogorov’s Extension Theorem
- circle23.12: The Fréchet-Nikodym Metric
- Chapter 24: Signed, Complex, and Vector Measures
- Chapter 25: Radon Measures
- Chapter 26: Measurable Functions
- Chapter 27: The Lebesgue Integral
- Chapter 28: The Bochner Integral
- Chapter 29: Integration on Locally Compact Groups
- circleChapter 30: Notations
- Chapter 22: Set Systems
- Part 5: Calculus
- Chapter 31: Differential Calculus
- circle31.1: Derivatives and Remainders
- circle31.2: Differentiation With Respect to Sets
- circle31.3: The Mean Value Theorem
- circle31.4: Higher Derivatives
- circle31.5: Taylor’s Formula
- circle31.6: Partial Derivatives
- circle31.7: Power Series
- circle31.8: Inverse Mappings
- circle31.9: Derivatives on $\mathbb{R}^{n}$
- Chapter 32: Complex Analysis
- circleChapter 33: Notations
- Chapter 31: Differential Calculus
- Part 6: Operator Algebras
- Chapter 34: Banach Algebras
- Chapter 35: $C^{*}$-Algebras
- circle35.1: Involutions
- circle35.2: Subalgebras
- circle35.3: Unitary Elements
- circle35.4: Self-Adjoint Elements
- circle35.5: *-Homomorphisms
- circle35.6: The Gelfand-Naimark Theorem
- circle35.7: The Continuous Functional Calculus
- circle35.8: Order Structures of $C^{*}$-Algebras
- circle35.9: Positive Linear Functionals
- circle35.10: States
- circle35.11: The GNS Construction
- Chapter 36: Examples of Operator Algebras
- circle36.1: The Matrix Algebra
- circle36.2: $B(H)$
- circle36.3: $C_{0}(X)$
- circle36.4: The Hardy Space
- circle36.5: The Disk Algebra
- circle36.6: $\ell^{1}(\integer)$
- circle36.7: $BC(X)$
- circleChapter 37: Notations
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