Analysis is the systematic study of real and complex-valued continuous functions.

Mathematical structures in analysis. Figure: Mathematical structures in analysis, $(X, \cdots)$: (A) topological and algebraic structures; (B) set-theoretic structure. Mathematical structures in (A) and (B) are independent but can be related via $L^2_\mu$.

Course Notes on Mathematical Analysis 1&2

Notes on Mathematical Analysis 3

Real number; function; continuity; convergence; differentiation; integration.

Course Notes on Complex Analysis

- Topological space: open set and closed set, continuous mapping;
- Vector Space: spaces with algebraic structure;
- Metric Space: convergence (of sequences);
- Normed Space (and Banach Space);
- Inner Product Space (and Hilbert Space);
- Special Operators;

Concepts:

Handouts:

- Unitary Transformations in L2(I, C) - Bochner's Theorem
- Distance to a subspace not attainable
- The Kernel Method
- Open Mapping Theorem
- Closed Subspaces with Zero Aperture

Vector calculus:

Curvilinear coordinates:

- Peter Lax, 1966. Functional Analysis;
- Michael Reed and Barry Simon, 1972. Functional Analysis;
- Erwin Kreyszig, 1978. Introductory Functional Analysis with Applications;