# Contents

## Idea

In mathematics, analysis usually refers to any of a broad family of fields that deals with a general theory of limits in the sense of convergence of sequences (or more generally of nets), particularly those fields that pursue developments that originated in “the calculus”, i.e., the theory of differentiation (differential calculus) and integration (integral calculus) of real and complex-valued functions. The classical foundation of this general subject is usually based on the idea that the real number system is describable as the (essentially unique) complete ordered field, or more generally on the concept of metric spaces. Their distance functions allow to formalize concepts like continuity and convergence in terms of existence of sufficiently small open balls. Many concepts of this “epsilontic analysis” have equivalent formulations in terms of simple combinatorics of open subsets with respect to the metric topology of metric spaces, and this way the field of analysis has a large overlap with the field of topology, this is particularly true for functional analysis and the theory of topological vector spaces.

Analysis can also refer to other responses to the problem of founding these developments, especially “infinitesimal analysis” which admits infinitesimal quantities not found in the classical real number system and which takes various forms, for example the nonstandard analysis first introduced by Abraham Robinson, or “synthetic differential analysis” whose rigorous foundations were largely introduced by William Lawvere and other category theorists who, following the example of Alexander Grothendieck, consider nilpotent infinitesimals (instead of invertible ones à la Robinson) as a basis for understanding differentiation.

### On mainstream analysis

Some of the $n$lab entries related to mathematical analysis include metric space, normed vector space, metric topology, sequence, net, convergence,

functional analysis, harmonic analysis, complex analysis, Weierstrass preparation theorem, several complex variables, Fourier transform, Pontrjagin dual, differential geometry, Legendre polynomial, dilogarithm, Hilbert space, Banach space, Banach algebra, topological vector space, locally convex space, operator algebras, Gelfand spectrum, measure space, measurable function, Lebesgue space, Sobolev space, bounded operator, compact operator, Fredholm operator, distribution (generalized function), hyperfunction.spectral theory, integral, integration…and a book entry Handbook of analysis and its foundations. Many of the basic notions used in analysis courses are described in $n$lab in the more general topological context if they belong there, e.g. compact space, continuous map, compact-open topology and so on. Many of the aspects of analytic geometry are treated in terms of Riemann surfaces, monodromy, local systems and so on.

### On foundations

Alternative foundations, especially constructive and those using topos theory, are of traditional interest to the category theory community. For example the synthetic differential geometry of Lawvere and Kock (more in next paragraph) and the nonstandard analysis of Robinson, and its variant, internal set theory of Nelson are some of the principal examples. See also Fermat theory, natural numbers object, infinitesimal number etc. Many statements are about the versions without the axiom of choice and so on; we like to state clean and minimal conditions when possible.

### On smoothness and generalized Lie theory

Various smoothness concepts in geometry, rarely studied in standard courses of analysis, but sometimes relevant, were studied to fair extent (and sometimes with innovations) in the $n$lab. These smoothness concepts are built using some primitive notions in rather generalized (often categorical) setups: Kähler differential, differential form, tangent space, jet bundles, resolution of diagonal, infinitesimal object, microlinear space, generalized smooth algebra, tangent category, cotangent complex as defining ingredients of various notions of smoothness and smooth spaces. Main framework to systematize in geometry similar notion studied in $n$lab is synthetic differential geometry but many other examples are also represented. Let us mention generalized smooth space, stratifold, Frölicher space, and some graded and super analogues (supermanifold, NQ-supermanifold, integration over supermanifolds); some concepts of smoothness are rather algebraic, e.g. formal smoothness of Grothendieck; see also algebraic approaches to differential calculus. Special attention in $n$lab has been paid to smooth group like objects like Lie group, Lie groupoid and their superanalogues and categorifications, as well as to their tangent structures like Lie algebroids and their interrelations (Lie theory: integration, Lie integration).

### On geometric function theory and quantization

Some other entries are related to the conceptual and categorical understanding of Feynman path integral, however so far from physical, conceptual and formal point of view only (and not of analytic theory). This is closely related to understanding various higher categorical spaces of sections in geometry and in study of sigma-models in physics. This is here called geometric function theory (cf. space and quantity, geometric quantization…).

### On quantization and the geometry of differential operators

Very relevant for quantization is also the geometric study of differential operators (see D-geometry, diffiety) and distributions (cf. microlocal analysis), by analysis of oscillating integrals (semiclassical approximation), symplectic geometry (esp. the geometry of lagrangian submanifolds which could often be viewed as quantum points) etc. Some of the topological properties of differential operators are studied in index theory, where special role have so called Dirac operators. Sometimes it is possible or even useful to avoid fine analysis by using the algebraic approaches to differential calculus and differential operators, what also makes possible some noncommutative analogues.

## References

### General

Textbooks accounts:

Discussion of the history, amplifying its roots all the way back in Zeno's paradoxes of motion is in

• Carl Benjamin Boyer, The history of the Calculus and its conceptual development, Dover 1949

### Constructive analysis

The formulation of analysis in constructive mathematics, hence constructive analysis, was maybe initiated in

• Errett Bishop, Foundations of constructive analysis. McGraw-Hill, (1967)

together with the basic notion of Bishop set/setoid. Implementations of constructive real number analysis in type theory implemented in Coq are discussed in

• R. O’Connor, A Monadic, Functional Implementation of Real Numbers. MSCS, 17(1):129-159, 2007 (arXiv:0605058)

• R. O’Connor, Certified exact transcendental real number computation in Coq, In TPHOLs 2008, LNCS 5170, pages 246–261, 2008.

• R. O’Connor, Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory, PhD thesis, Radboud University Nijmegen, 2009.

• Robbert Krebbers, Bas Spitters, Type classes for efficient exact real arithmetic in Coq (arXiv:1106.3448)