Math 540: Linear Analysis

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Catalog Information

Linear Analysis.

(3:3:0)

Recommended

Math 342 or equivalent.

Description

Normed vector spaces and linear maps between them.

Desired Learning Outcomes

The course is designed to cover elementary abstract linear functional analysis. "Elementary" means that methods dependent on complex analysis or measure-theoretic integration are not core topics. "Abstract" means that applications to specific function spaces are not core topics.

Prerequisites

The official prerequisite is Math 316. What's important is that incoming students be familiar with linear algebra and metric spaces and be mathematically mature.

Minimal learning outcomes

Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.

1. Normed spaces
• Basics
• Banach spaces
• Special linear operators
• Continuous/bounded
• Compact
• Finite rank
• Duality
• Dual spaces
• Their completeness
• Adjoints of bounded linear operators
• Second duals
• Reflexivity
• Weak and weak-star topologies
• Banach-Alaoglu theorem
• Structure
• Hamel and Schauder bases
• Biorthogonal systems
• Separability
• Direct sums
• Quotient spaces
• Finite-dimensional spaces
• Equivalence of all norms
• Completeness
• Continuity of all linear operators
• Characterization: unit ball is compact
• Fundamental theorems
• Baire category theorem
• Hahn-Banach extension theorem
• Banach-Steinhaus theorem
• Open mapping theorem
• Closed graph theorem
• Bounded inverse theorem
2. Inner product spaces
• Basics
• Hilbert spaces
• Special linear operators
• Self-adjoint
• Unitary
• Normal
• Orthogonal projections
• Hilbert-Schmidt operators
• Structure
• Orthogonality
• Complements and direct sums
• Bases
• Representation theorems
• Riesz-Frechet theorem
• Lax-Milgram theorem
• Abstract Fourier theory
• Riesz-Fischer theorem
• Bessel’s inequality
• Parseval’s identities
3. Spectral theory
• Banach algebras
• Bounded operators on Banach spaces
• Gelfand’s spectral-radius formula
• Compact operators on Banach spaces
• Riesz-Schauder theory including Fredholm Alternative
• Compact normal operators on Hilbert spaces
• Compact self-adjoint operators on Hilbert spaces

Additional topics

While the focus of the course is on abstract theory, this theory should probably be motivated and illustrated with appropriate concrete examples.

Courses for which this course is prerequisite

This course would be a prerequisite for the Math 640. Indirectly (through the Math 640), this course would be a prerequisite for Math 647.