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.