Analysis

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 $L^2_\mu$.

Mathematical Analysis

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.

Real Analysis

Set Theory and Measure Space

Complex Analysis

Course Notes on Complex Analysis

Functional Analysis

1. Metric Space: subspace, product space
2. Mapping: functional, operator/transform
3. Continuity (of functions)
4. Convergence (of sequences): uniform convergence
5. Topology: open set and closed set
6. Vector Space: spaces with algebraic structure.
7. Normed Space (and Banach Space)
8. Inner Product Space (and Hilbert Space)
9. Special Operators

Concepts:

• Equivalence Concepts
• Principle of Duality

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

Misc

Vector calculus:

• Vector derivatives & common coordinate systems
• Vector calculus identities

Curvilinear coordinates:

• Common coordinate systems
• Curvilinear coordinates
• Rotating coordinates

References

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

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🏷 Category=Analysis