# Math 540: Linear Analysis

## Contents

## Catalog Information

### Title

Linear Analysis.

### (Credit Hours:Lecture Hours:Lab Hours)

(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 342. 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.

- 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

- Dual spaces
- 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

- Basics
- 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

- Orthogonality

- Basics
- 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

### Textbooks

Possible textbooks for this course include (but are not limited to):

- Bela Bollobas,
*Linear Analysis (2nd edition)*, Cambridge, 2008.

### 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 is recommended for Math 640. Indirectly (through the Math 640), this course will possibly become be a prerequisite for Math 647.