Analysis

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

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

Mathematical Analysis

Course Notes on Mathematical Analysis 1&2

Notes on Mathematical Analysis 3

1. Real number
2. Function
3. Continuity
4. Convergence
5. Differentiation
6. 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

Lebesgue Measure and Integration

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

Appendices

Vector calculus:

• Vector derivatives & common coordinate systems
• Vector calculus identities

Curvilinear coordinates:

• Common coordinate systems
• Curvilinear coordinates
• Rotating coordinates

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🏷 Category=Analysis