Analysis

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

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

Real Analysis

Set Theory & Measure Space

Lebesgue Measure & 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. Linear Space
7. Normed Linear & Banach Space
8. Inner Product & Hilbert Space
9. Special Operators

Concepts:

• Equivalence Concepts
• Duality 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

Appendices

Vector calculus:

• Vector derivatives & common coordinate systems
• Vector calculus identities

Curvilinear coordinates:

• Common coordinate systems
• Curvilinear coordinates
• Rotating coordinates