# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 202: Functional Analysis

**Approved:**2007-09-01, Roman Vershynin

**Units/Lecture:**

Winter, alternate years; 4 units; lecture/term paper

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Eidelman, Yuli; Milman, Vitali; Tsolomitis, Antonis Functional Analysis. An introduction. Graduate Studies in Mathematics, 66. American Mathematical Society, Providence, RI, 2004. ($58)

Search by ISBN on Amazon: 0-8218-3646-3

Search by ISBN on Amazon: 0-8218-3646-3

**Prerequisites:**

MAT 201AB or consent of instructor.

**Course Description:**

Fundamental theorems: Hahn-Banach, Open mapping, Closed graph, Banach-Steinhaus, and Krein-Milman. Subspaces and quotient spaces. Projections. Weak and weak-star topologies. Compact and adjoint operators in Banach spaces. Fredholm theory. Functions of operators. Spectral theory of self-adjoint operators.

Applications: subspaces and quotient spaces, projections in Banach spaces, separation of convex sets in linear vector spaces, weak and weak* topologies. Compact operators, adjoint operators in Banach spaces. Fredholm theory of compact operators. Functions of operators. Spectral theory of self-adjoint bounded operators on Hilbert space.

Applications: subspaces and quotient spaces, projections in Banach spaces, separation of convex sets in linear vector spaces, weak and weak* topologies. Compact operators, adjoint operators in Banach spaces. Fredholm theory of compact operators. Functions of operators. Spectral theory of self-adjoint bounded operators on Hilbert space.

**Suggested Schedule:**

Lectures | Sections | Topics/Comments |
---|---|---|

This course covers Chapters 9, 4.4, 5, 7 (in this order). Most of Chapters 1, 2 are covered in the prerequisite course 201. Chapter 7 (Functions of operators, spectral decomposition) can be replaced by Chapter 10 (Banach algebras) or Chapter 11 (Unbounded self-adjoint operators). | ||

9.1 | Baire category theorem | |

9.2 | Open mapping theorem. Inverse mapping theorem. Equivalent norms on a Banach space. | |

9.3 | Closed graph theorem. Example: Hoermander's bound on the sup-norm of the derivative. Projections in linear spaces (review 6.4a). Complemented subspaces. Boundedness of projections onto closed subspaces (Theorem 9.3.5). | |

9.4 | Banach-Steinhaus theorem and its consequences. Applications: boundedness of self-adjoint operators (Theorem 9.4.8), integration formulas (Polya's theorem), divergent Fourier series of a continuous function. Some of these applications can be skipped. | |

[9.5 | Bases in Banach spaces - SKIP] | |

9.6 | Hahn-Banach theorem. Review Zorn's lemma before the proof. Immediate consequences: extensions of functionals, supporting functionals; the dual space separates the points. Annihilators of subspaces (Corollaries 3.1.7 and 3.1.8). Quotient spaces: definition (pp.5-6), quotient norm (Lemma 1.4.1), completeness (Exercise 1.6.24). Duality between subspaces and quotient spaces (Corollary 9.6.10). Reflexive spaces. Reflexivity of subspaces (Corollary 9.6.11). Reflexivity of quotient spaces can be given as an exercise. Functionals on a reflexive space attain their norm. Converse statement: James' Theorem 9.8.2 (without proof). | |

9.7 | Separation of convex sets. Exercise 9.10.30 can be covered in the lecture. Weak and weak* topologies. Mazur's lemma. Strong and weak closedness are equivalent. Alaoglu's theorem. [Skip the rest of Section 9.7]. | |

9.8 | Eberlein-Schmulian theorem. May be skipped. The proof uses that X* separable implies X separable - a statement that has to be proved separately. | |

9.9 | Krein-Milman theorem for weakly compact sets. Thus only prove Theorem 9.9.1 for THE weak topology. In the proof, one can use the separation theorem (Corollary 9.7.4) instead of Theorem 9.7.11. Applications (may skip some): extreme points of the set of doubly stochastic matrices (Birkhoff's theorem), of the set of unitary operators, of the balls of C[0,1], L1[0,1]. Assign or work out exercises 9.10.36. Compact operators (review Section 4.3). 4.4. Adjoint operators. Compactness of the adjoint. | |

5.1 | Spectrum of operators on a Banach space. Classification of the spectrum. Spectrum of compact operators (Propositions 5.2.3 and 5.2.4). Fredholm theory (the rest of Section 5.2). Self-adjoint operators. Review the spectral theory of compact operators (Section 6.2a). 6.3. Order in the space of self-adjoint operators. The square root of an operator (Lemma 6.3.5). | |

7. | Functions of operators. 7.1. The spectral decomposition. The spectral integral. The spectral theorem for self-adjoint operators on Hilbert space (Hilbert's Theorem 7.2.1). Characterization of the spectrum in terms of the spectral family (Theorems 7.3.1 and 7.3.2). | |

[7.4 | Simple spectrum - MAY SKIP] |

**Assessment:**

GRADING NOTE: The textbook includes solutions to all exercises. The instructor may choose to give homework and/or exams from other texts, such as W. Rudin, Functional Analysis, or J. B. Conway, Functional Analysis.