Analysis

Mathematical 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 structures in analysis:

• Measure Space: integration;
• Topological space: open set and closed set, continuous mapping, differentiation;
  • Metric Space: convergent sequence;
• Vector Space: basis, linear operator;
• Normed Space (and Banach Space): approximation;
• Inner Product Space (and Hilbert Space): orthogonality, generalized Fourier series;
• Special Linear Operators;

Concepts:

• Equivalence Concepts
• Principle of Duality

Notes:

• Course Notes on Mathematical Analysis 1&2
• Notes on Mathematical Analysis 3
• Course Notes on Complex Analysis

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

Concepts

Real function $f: \mathbb{R} \mapsto \mathbb{R}$ is a mapping between subsets of the real line. Real-valued function $f: X \mapsto \mathbb{R}$ is a mapping whose codomain is the real line. Function of a real variable $f: \mathbb{R} \mapsto X$ is a mapping whose domain is the real line.

Support $\mathrm{supp}(f)$ of a real-valued function $f: X \mapsto \mathbb{R}$ on a topological space $(X, \mathcal{T})$ is the closure of the subspace where $f$ is not zero: $\mathrm{supp}(f) := \overline{\{x : f(x) \ne 0\}}$.

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