# Math 540: Linear Analysis

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Linear Analysis.

(3:3:0)

### 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
• 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
• Compact operators on Banach spaces
• Riesz-Schauder theory including Fredholm Alternative
• Compact normal operators on Hilbert spaces
• Compact self-adjoint operators on Hilbert spaces