1.Introduction 2.Categories 3.The Category of Groups 4.Subgroups 5.Normal Subgroups 6.Homomorphisms 7.Direct Products and Sums of Groups 8.Relations 9.The Category of Vector Spaces 10.Subspaces 11.Linear Mappings; Direct Products and Sums 12.From Real to Complex Vector Spaces and Back 13.Duals 14.Multilinear Mappings; Tensor Products 15.Example: Minkowski Vector Space 16.Example: The Lorentz Group 17.Functors 18.The Category of Associative Algebras 19.The Category of Lie Algebras 20.Example: The Algebra of Observables 21.Example: Fock Vector Space 22.Representations: General Theory 23.Representations on Vector Spaces 24.The Algebraic Categories: Summary 25.Subsets and Mappings 26.Topological Spaces 27.Continuous Mappings 28.The Category of Topological Spaces 29.Nets 30.Compactness 31.The Compact-Open Topology 32.Connectedness 33.Example: Dynamical Systems 34.Homotopy 35.Homology 36.Homology: Relation to Homotopy 37.The Homology Functors 38.Uniform Spaces 39.The Completion of a Uniform Space 40.Topological Groups 41.Topological Vector Spaces 42.Categories: Summary 43.Measure Spaces 44.Constructing Measure Spaces 45.Measurable Functions 46.Integrals 47.Distributions 48.Hilbert Spaces 49.Bounded Operators 50.The Spectrum of a Bounded Operator
51.The Spectral Theorem: Finite-dimensional Case 52.Continuous Functions of a Hermitian Operator 53.Other Functions of a Hermitian Operator 54.The Spectral Theorem 55.Operators (Not Necessarily Bounded) 56.Self-Adjoint Operators Index of Defined Terms
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