Functional Analysis | Mathematics

sources:

• text: Spivak - Calculus

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

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

• 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

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

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

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

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

Functional analysis extends linear algebra to infinite dimensions, studying spaces of functions rather than finite tuples of numbers. A normed space adds a notion of length to a vector space; completeness ensures that Cauchy sequences converge, producing Banach spaces. When the norm comes from an inner product, we get Hilbert spaces, the natural setting for quantum mechanics. Bounded linear operators are the continuous transformations between these spaces. The great theorems of functional analysis, including Hahn-Banach and the open mapping theorem, reveal the surprisingly rigid structure that completeness imposes. The key insight is that many problems in analysis, differential equations, and quantum theory become statements about geometry in function space.

Mistake 1: Confusing Banach spaces with Hilbert spaces Not every complete normed space is a Hilbert space. A Hilbert space requires the norm to come from an inner product, which is equivalent to the parallelogram law holding. The space $L^1$ is a Banach space but not a Hilbert space because its norm does not satisfy the parallelogram identity.

Mistake 2: Assuming all bounded linear operators are compact Boundedness does not imply compactness. The identity operator on an infinite-dimensional Hilbert space is bounded but not compact. Compact operators map bounded sets to precompact sets, which is a strictly stronger condition than continuity in infinite dimensions.

Mistake 3: Forgetting that duality in $L^p$ spaces requires $1 \leq p < \infty$ The dual of $L^p(\mu)$ is $L^q(\mu)$ where $1/p + 1/q = 1$, but this fails for $p = \infty$. The dual of $L^1$ is $L^\infty$, but the dual of $L^\infty$ is strictly larger than $L^1$ when the measure space is not finite. Students often blindly apply the $L^p$ duality formula to $p = \infty$. 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

• Linear Algebra: Vector spaces and linear operators.

• Real Analysis: Foundational analysis extended by functional analysis.

• Measure Theory: L^p spaces and measure-theoretic foundations.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism