Jerry's Digital Garden

Contents

• keyboard_arrow_downPart 1: General Tools
  • keyboard_arrow_rightChapter 1: Category Theory
    • circle1.1: Categories and Functors
    • circle1.2: Universal Construction
    • circle1.3: Universal Constructions for Modules
  • circleChapter 2: Gluing Lemmas
  • keyboard_arrow_rightChapter 3: Inequalities and Computations
    • circle3.1: Dyadic Rational Numbers
    • circle3.2: Product Inequalities
  • circleChapter 4: Notations
• keyboard_arrow_downPart 2: General Topology
  • keyboard_arrow_rightChapter 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
    • keyboard_arrow_right5.20: 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
  • keyboard_arrow_rightChapter 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
  • keyboard_arrow_rightChapter 7: Function Spaces
    • circle7.1: Ideals
    • circle7.2: Topologies With Respect to Ideals
    • circle7.3: The Uniform Topology
    • circle7.4: The Stone-Weierstrass Theorem
  • keyboard_arrow_rightChapter 8: Metric Spaces
    • circle8.1: Metrics
    • circle8.2: Distance Between Sets
  • keyboard_arrow_rightChapter 9: Topological Groups
    • circle9.1: Group Topologies
    • circle9.2: Coset Spaces and Factor Groups
  • circleChapter 10: Notations
• keyboard_arrow_downPart 3: Functional Analysis
  • keyboard_arrow_rightChapter 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
  • keyboard_arrow_rightChapter 12: Locally Convex Spaces
    • circle12.1: 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
    • keyboard_arrow_right12.8: 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
  • keyboard_arrow_rightChapter 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
  • keyboard_arrow_rightChapter 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
  • keyboard_arrow_rightChapter 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
  • keyboard_arrow_rightChapter 16: Order Structures
    • circle16.1: Vector Lattices
    • circle16.2: Banach Lattices
  • keyboard_arrow_rightChapter 17: Duality
    • circle17.1: Dual Systems
    • circle17.2: Polars
  • keyboard_arrow_rightChapter 18: Interpolation Spaces
    • circle18.1: Compatible Couples
    • circle18.2: The Complex Interpolation Method
  • circleChapter 19: Notations
• keyboard_arrow_downPart 4: Measure Theory and Integration
  • keyboard_arrow_rightChapter 20: Set Systems
    • circle20.1: Algebras
    • circle20.2: The Borel $\sigma$-Algebra
    • circle20.3: Lambda Systems
    • circle20.4: Elementary Families
    • circle20.5: Limits of Sets
  • keyboard_arrow_rightChapter 21: Positive Measures
    • circle21.1: Measures
    • circle21.2: Complete Measures
    • circle21.3: Semifinite Measures
    • circle21.4: $\sigma$-Finite Measures
    • circle21.5: Regular Measures
    • circle21.6: Carathéodory’s Extension Theorem
    • circle21.7: Lebesgue-Stieltjes Measures
    • circle21.8: Product Measures
    • circle21.9: Kolmogorov’s Extension Theorem
    • circle21.10: The Fréchet-Nikodym Metric
  • keyboard_arrow_rightChapter 22: Signed, Complex, and Vector Measures
    • circle22.1: Signed and Complex Measures
    • circle22.2: Vector Measures
    • circle22.3: Total Variation
    • circle22.4: Absolute Continuity
    • circle22.5: Mutually Singular
    • circle22.6: The Lebesgue-Radon-Nikodym Theorem
    • circle22.7: Spaces of Finite Measures
  • keyboard_arrow_rightChapter 23: Radon Measures
    • circle23.1: Radon Measures
    • circle23.2: The Riesz Representation Theorem
    • circle23.3: Dual of $C_{0}$
  • keyboard_arrow_rightChapter 24: Measurable Functions
    • circle24.1: Measurable Functions
    • circle24.2: Product $\sigma$-Algebras
    • circle24.3: Real-Valued Measurable Functions
    • circle24.4: Simple Functions
    • circle24.5: Measurable Functions into Metric Spaces
    • circle24.6: Convergence in Measure
  • keyboard_arrow_rightChapter 25: The Lebesgue Integral
    • circle25.1: Integration of Simple Functions
    • circle25.2: Integration of Non-Negative Functions
    • circle25.3: Integration of Complex-Valued Functions
  • keyboard_arrow_rightChapter 26: The Bochner Integral
    • circle26.1: Strongly Measurable Functions
    • keyboard_arrow_right26.2: The Bochner Integral
      • circle26.2.1: Vector Measure Version
  • keyboard_arrow_rightChapter 27: Integration on Locally Compact Groups
    • circle27.1: Locally Compact Groups
    • circle27.2: Haar Measures
  • circleChapter 28: Notations
• keyboard_arrow_downPart 5: Calculus
  • keyboard_arrow_rightChapter 29: Differential Calculus
    • circle29.1: Derivatives and Remainders
    • circle29.2: Differentiation With Respect to Sets
    • circle29.3: The Mean Value Theorem
    • circle29.4: Higher Derivatives
    • circle29.5: Taylor’s Formula
    • circle29.6: Partial Derivatives
    • circle29.7: Power Series
    • circle29.8: Inverse Mappings
    • circle29.9: Derivatives on $\mathbb{R}^{n}$
  • keyboard_arrow_rightChapter 30: Complex Analysis
    • circle30.1: Complex Differentiability
    • circle30.2: Zeroes of Holomorphic Functions
    • circle30.3: The Riemann Sphere
    • circle30.4: Spaces of Holomorphic Functions
    • circle30.5: The Complex Logarithm
    • circle30.6: Entire Functions
    • circle30.7: Runge’s Theorem
  • circleChapter 31: Notations
• keyboard_arrow_downPart 6: Operator Algebras
  • keyboard_arrow_rightChapter 32: Banach Algebras
    • circle32.1: Banach Algebras
    • circle32.2: Invertible Elements
    • circle32.3: The Index Group
    • circle32.4: Ideals
    • circle32.5: The Spectrum
    • circle32.6: The Holomorphic Functional Calculus
    • circle32.7: Multiplicative Functionals
    • circle32.8: The Gelfand Transform
  • keyboard_arrow_rightChapter 33: $C^{*}$-Algebras
    • circle33.1: Involutions
    • circle33.2: Self-Adjoint Elements
    • circle33.3: Order Structures of $C^{*}$-Algebras
  • keyboard_arrow_rightChapter 34: Examples of Operator Algebras
    • circle34.1: The Matrix Algebra
    • circle34.2: $B(H)$
    • circle34.3: $C_{0}(X)$
    • circle34.4: The Hardy Space
    • circle34.5: The Disk Algebra
    • circle34.6: $\ell^{1}(\integer)$
    • circle34.7: $BC(X)$
  • circleChapter 35: Notations