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:

Concepts:

Notes:

Handouts:

## 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:

Curvilinear coordinates:

## References

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