Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function. It also includes the theories of differentiation, integration and measure, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or more specifically "distance" (a metric space).
Proofs are arranged in the following sections:
- Metric Spaces
- Real Analysis
- Functional Analysis
- Harmonic Analysis
- Complex Analysis
- Differential Geometry
- p-adic Analysis
- Non-standard Analysis or model theory
- Numerical Analysis
Most of the standard proofs of analysis studied at the undergraduate level are in the metric spaces section. This is so because proofs of such results in real analysis, complex analysis and even in topology are similar to them.