Autumn 2005

MAT4340: Elementary Functional Analysis

MAT4340: Elementary Functional Analysis

Main book:

Y. Eidelman, V. Milman, A. Tsolomitis. Functional analysis. An introduction.

Additional literature:

W. Rudin. Functional analysis.

G. Pedersen. Analysis now.

A. N. Kolmogorov, S. V. Fomin. Elements of the theory of functions and functional analysis.

Lecture 1 (23.08)

Normed spaces. Quotient normed spaces.

Sections: 1.1-1.4.

Lecture 2 (25.08)

Completion of normed spaces. Banach spaces. Hilbert spaces.

Sections: 1.5, 2.1a.

Exercises: 1, 11, 12, 13, 20, 21 in Chapter 1, and these five.

Lecture 3 (06.09)

Bessel's inequality and Parseval's identity. Existence of orthonormal bases.

Sections: 2.1b-2.1d.

Lecture 4 (08.09)

Examples of orthonormal bases. Convex sets.

Sections: 2.1e, 2.2b.

Exercises: 1 in Chapter 2, and these four.

Lecture 5 (13.09)

Orthogonal projections.

Sections: 2.2a, 2.2c.

Lecture 6 (15.09)

Linear functionals. The dual space. Examples of dual spaces.

Sections: 2.3a-2.3c, 3.2.

Exercises: 11, 28, 29 in Chapter 2, and these four.

Lecture 7 (20.09)

Examples of dual spaces. The Hahn-Banach theorem.

Sections: 3.1, 3.2.

Lecture 8 (22.09)

Corollaries of the Hahn-Banach theorem. Applications of the Hahn-Banach theorem: cardinality of a Hamel basis in a Banach space, Banach limits.

Sections: 3.1.

Exercises: 24 in Chapter 2, and these seven.

Lecture 9 (27.09)

Bounded linear operators.

Sections: 4.1.

Lecture 10 (29.09)

Examples of bounded operators. Precompact sets. Compact operators.

Sections: 4.2, 4.3a.

Exercises: 19 in Chapter 4, and these five.

Lecture 11 (04.10)

Finite rank operators.

Sections: 4.5.

Lecture 12 (06.10)

Approximation of compact operators by operators of finite rank. Dual operators.

Sections: 4.4, 4.5a.

Exercises: these three.

Lecture 13 (11.10)

Different types of operator convergences.

Sections: 4.6.

Lecture 14 (13.10)

Invertible operators. The spectrum of an operator. Spectral theory of compact operators: point spectrum.

Sections: 4.7, 5.1, 5.2.

Lecture 15 (18.10)

Spectral theory of compact operators: continuous spectrum.

Sections: 5.2.

Lecture 16 (20.10)

Spectral theory of compact operators: residual spectrum. The Fredholm alternative.

Sections: 5.2.

Exercises: 3, 4, 11 in Chapter 5, and these five.

Lecture 17 (25.10)

Adjoint operators.

Sections: 4.4, 6.1.

Lecture 18 (27.10)

Spectrum of a self-adjoint operator. Compact self-adjoint operators.

Sections: 6.1, 6.2a.

Exercises: these three.

Lecture 19 (01.11)

Minimax principle.

Sections: 6.2b.

Lecture 20 (8.11)

The second Hilbert-Schmidt theorem. Order on the space of self-adjoint operators.

Sections: 6.2c, 6.3.

Lecture 21 (10.11)

Polar decomposition. Functional calculus for continuous functions.

Sections: 6.3, 7.0.

Exercises: these two.

Lecture 22 (15.11)

Functional calculus for discontinuous functions.

Sections: 7.0.

Lecture 23 (17.11)

Functional calculus for discontinuous functions. Orthoprojections. Spectral family. Spectral integral.

Sections: 6.4, 7.0, 7.1.

Lecture 24 (22.11)

Spectral integral.

Sections: 7.1b.

Lecture 25 (24.11)

Spectral decomposition of self-adjoint operators.

Sections: 7.2.

Exercises: these three.