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

Figure: Mathematical structures in analysis, $(X, \cdots)$.

Course Notes on Mathematical Analysis 1&2

Notes on Mathematical Analysis 3

- Real number
- Function
- Continuity
- Convergence
- Differentiation
- Integration

A **sequence** is an ordered set of mathematical objects, $\{a_i\}_{i=1}^N$.
A **series** is an infinite sequence of partial sums, $\{s_n\}_{n=1}^\infty$,
$s_n = \sum_{i=1}^n a_i$; $a_n$ is called the $n$**-th term** of the series and
$s_n$ is called its **partial sum of order** $n$.
The study of series is equivalent to the study of sequences.

Lebesgue Measure and Integration

Course Notes on Complex Analysis

- Metric Space: subspace, product space
- Mapping: functional, operator/transform
- Continuity (of functions)
- Convergence (of sequences): uniform convergence
- Topology: open set and closed set
- Vector Space: spaces with algebraic structure.
- 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: