Unit 3117: Functional Analysis

1. Introduction to Functional Analysis

• Definition of functions
• Domains and ranges
• Operations on functions (addition, multiplication, composition)
• Basic examples and motivation for functional analysis

2. Metric Spaces and Normed Spaces

2.1 Metric Spaces

• Definition of a metric
• Examples of metric spaces
• Open and closed sets in metric spaces
• Convergence and continuity

2.2 Normed Spaces

• Definition of norm
• Relation between norm and metric
• Examples of normed spaces
• Open and closed sets in normed spaces
• Convergence and completeness

3. Banach Spaces and Hilbert Spaces

3.1 Banach Spaces

• Definition and examples
• Completeness in normed spaces
• Basic properties
• Basis in Banach spaces

3.2 Hilbert Spaces

• Inner product spaces
• Definition of Hilbert spaces
• Orthogonality and orthogonal projections
• Orthonormal bases
• The parallelogram law and polarization identity

4. Linear Operators and Functionals

4.1 Linear Operators

• Definition and examples
• Bounded vs unbounded operators
• Operator norm

4.2 Adjoint Operators

• Definition and properties
• Examples in Hilbert spaces

4.3 Compact Operators

• Definition and characterization
• Examples and spectral properties

4.4 Spectrum of Operators

• Definition of spectrum
• Point, continuous, and residual spectrum
• Spectral radius and spectral theorem overview

5. Distributions and Sobolev Spaces

5.1 Distributions

• Motivation and definition
• Test function spaces
• Operations on distributions

5.2 Sobolev Spaces

• Definition via weak derivatives
• Examples and properties
• Sobolev embedding theorems
• Trace theorems

6. Fourier Analysis

6.1 Fourier Series

• Definition and convergence
• Examples and applications

6.2 Fourier Transforms

• Definition and properties
• Plancherel theorem
• Convolution theorem
• Fourier inversion formula

7. Applications of Functional Analysis

• Applications in physics: quantum mechanics, PDEs
• Applications in engineering: signal processing, control theory
• Applications in economics: optimization, equilibrium theory
• Applications in applied mathematics and data science

Summary

• Recap of key concepts
• Integration of topics
• Discussion of further directions