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# Functional Analysis

**Course Description:**

This course can be thought of as a continuation of Mathematical Anmalysis III. Functional analysis a central pillar of modern analysis, and we will cover its foundations.

The main emphasis will be on the study of the properties of bounded linear maps between topological linear spaces of various kinds. This provides the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus. It also has a very close relation to measure and integration theory.

**Course Contents:**

I. Banach and Hilbert spaces

- Linear spaces. Subspaces and quotient spaces. Linear operators
- Normed spaces. Examples: L_p, C(K), l_p
- Banach spaces. Series. Examples (same)
- Subspaces and quotient spaces
- Hilbert spaces. Orthogonal projections, decompositions
- Orthogonal bases. Fourier series
- Applications: ergodic theorems

II. Linear functionals

- Definition and examples. Norm
- Hahn-Banach Theorem (analytic and geometric versions)
- Dual spaces
- General form of linear functionals in concrete Banach spaces, and in Hilbert space
- Applications: Banach limit, invariant means

III. Linear operators in Banach spaces

- Defintion and examples. Norm. Isomorphisms and isometries. Continuity and boundedness
- Space of bounded linear operators
- Extension of operators
- Adjoint operators
- Duality between subspaces and quotient spaces

IV. Main principles of Functional Analysis

- Open mapping theorem
- Inverse mappings, isomorphisms
- Isomorphism of all finite dimensional Banach spaces. Isometry of all Hilbert spaces.
- Closed graph theorem
- Banach-Steinhaus theorem. Applications in Fourier analysis
- Compact sets in Banach spaces. Compactness criteria in concrete spaces
- Weak and weak* convergence and topologies. Second dual
- Alaoglu theorem
- Krein-Milman theorem

V. Elements of spectral theory

- Definition and properties of spectrum
- Compact operators. Structure of their spectrum
- Selfadjoint operators in Hilbert space
- Spectral theorem for compact selfadjoint operators
- Hilbert-Schmidt operators
- Projections
- Unitary operators. Polar decomposition

Measure Theory, Point Set Topology, Advanced Linear Algebra

Projects 20%

Midterm 30%

Final 50%

N/A

Each week, Sunday and Tuesday

From 8:00 am to 10:00 am

Class # 8 in the Mathematics Building