1. Linear Algebra. - 1. 1 Vector Spaces and Duality. - 1. 2 Direct Sums and Quotients. - 1. 3 Inner-Product Spaces. - 1. 4 The Spectral Theorem. - 1. 5 Fields and Field Extensions. - 1. 6 Existence of Bases for Infinite-Dimensional Spaces. - 1. 7 Notes. - 1. 8 Exercises. - 2. Algebras. - 2. 1 Algebrai c Structures. - 2. 2 Algebras with a Prescribed Basis. - 2. 3 Algebras of Linear Transformations. - 2. 4 Inversion and Spectra. - 2. 5 Division Algebras and Other Simple Algebras. - 2. 6 Notes. - 2. 7 Exercises. - 3. Invariant Subspaces. - 3. 1 The Invariant-Subspace Lattice. - 3. 2 Idempotents and Projections. - 3. 3 Existence of Invariant Subspaces. - 3. 4 Representations and Left Ideals. - 3. 5 Functional Calculus and Polar Decomposition. - 3. 6 Notes. - 3. 7 Exercises. - 4. Semisimple Algebras. - 4. 1 Nilpotent Algebras and the Nil Radical. - 4. 2 Structure of Semisimple Algebras. - 4. 3 Structure of Simple Algebras. - 4. 4 Isomorphism Classes of Semisimple Algebras. - 4. 5 Notes. - 4. 6 Exercises. - 5. Operator Algebras. - 5. 1 Von Neumann Algebras. - 5. 2 Real and Complex Involutive Algebras. - 5. 3 Representation of Operator Algebras. - 5. 4 Wedderburn Theorems for Operator Algebras. - 5. 5 C*-Algebras. - 5. 5 Notes. - 5. 7 Exercises. - 6. Tensor Products. - 6. 1 Free Vector Spaces. - 6. 2 Tensor Products of Vector Spaces. - 6. 3 Tensor Products of Algebras. - 6. 4 Tensor Products of Operator Algebras. - 6. 5 Notes. - 6. 6 Exercises. - References. Language: English