Functional Analysis

Contents

1. Normed Spaces and Banach Spaces
2. Inner Product Spaces and Hilbert Spaces
3. Bounded Linear Operators
4. The Fundamental Theorems
5. Compact Operators
6. Weak and Weak* Convergence
7. Applications
8. Historical Context
9. Summary of Key Theorems

Overview

University-level functional analysis notes covering normed spaces, Hilbert spaces, and operators.

Topics Covered

• Normed and Banach Spaces: Definitions, completeness, examples
• Hilbert Spaces: Inner products, orthogonality, projections
• Bounded Linear Operators: Continuity, duality, spectrum
• Fundamental Theorems: Hahn-Banach, Open Mapping, Closed Graph

Prerequisites

• Real analysis (sequences, series, continuity, differentiation)
• Linear algebra (vector spaces, linear maps)
• Basic topology (open sets, compactness)
• Mathematical proofs and logic

How to Use These Notes

Start with normed spaces to build foundational knowledge, then progress to Hilbert spaces and operators. Each section includes worked examples and practice problems.

Navigation

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

• Detailed explanations of key concepts
• Worked examples with step-by-step solutions
• Practice problems with answers
• Common pitfalls and how to avoid them
• Connections to other areas of mathematics

Study Tips

1. Master the definitions: Functional analysis requires precise understanding of norms and inner products
2. Practise proofs: Learn to write clear, rigorous proofs
3. Draw diagrams: Visualise function spaces and operators
4. Learn standard examples: Know the properties of common spaces (l-p spaces, L-p spaces)
5. Connect to analysis: Relate functional analysis to real analysis and PDEs