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\section*{analysis}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{entries_related_to_analysis}{Entries related to analysis}\dotfill \pageref*{entries_related_to_analysis} \linebreak
\noindent\hyperlink{on_mainstream_analysis}{On mainstream analysis}\dotfill \pageref*{on_mainstream_analysis} \linebreak
\noindent\hyperlink{on_foundations}{On foundations}\dotfill \pageref*{on_foundations} \linebreak
\noindent\hyperlink{on_smoothness_and_generalized_lie_theory}{On smoothness and generalized Lie theory}\dotfill \pageref*{on_smoothness_and_generalized_lie_theory} \linebreak
\noindent\hyperlink{on_geometric_function_theory_and_quantization}{On geometric function theory and quantization}\dotfill \pageref*{on_geometric_function_theory_and_quantization} \linebreak
\noindent\hyperlink{on_quantization_and_the_geometry_of_differential_operators}{On quantization and the geometry of differential operators}\dotfill \pageref*{on_quantization_and_the_geometry_of_differential_operators} \linebreak
\noindent\hyperlink{on_contructivism_and_computable_analysis}{On contructivism and computable analysis}\dotfill \pageref*{on_contructivism_and_computable_analysis} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{ReferencesGeneral}{General}\dotfill \pageref*{ReferencesGeneral} \linebreak
\noindent\hyperlink{ReferencesConstructiveAnalysis}{Constructive analysis}\dotfill \pageref*{ReferencesConstructiveAnalysis} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

In [[mathematics]], \emph{analysis} usually refers to any of a broad family of fields that deals with a general theory of \emph{[[limit of a sequence|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 numbers|real]] and [[complex numbers|complex]]-valued [[functions]]. The classical foundation of this general subject is usually based on the idea that the [[real numbers|real number system]] is describable as the (essentially unique) [[complete space|complete]] [[ordered field]], or more generally on the concept of [[metric spaces]]. Their [[distance]] functions allow to formalize concepts like [[continuous functions|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 \emph{[[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 geometry|synthetic differential analysis]]'' whose rigorous foundations were largely introduced by [[William Lawvere]] and other [[category theory|category theorists]] who, following the example of [[Alexander Grothendieck]], consider [[nilpotent infinitesimals]] (instead of invertible ones \`a{} la Robinson) as a basis for understanding [[differentiation]].

\hypertarget{entries_related_to_analysis}{}\subsection*{{Entries related to analysis}}\label{entries_related_to_analysis}

\hypertarget{on_mainstream_analysis}{}\subsubsection*{{On mainstream analysis}}\label{on_mainstream_analysis}

Some of the $n$lab entries related to \textbf{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]]\ldots{}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 [[topology|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 system]]s and so on.

\hypertarget{on_foundations}{}\subsubsection*{{On foundations}}\label{on_foundations}

Alternative [[foundations]], especially [[constructive mathematics|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.

\hypertarget{on_smoothness_and_generalized_lie_theory}{}\subsubsection*{{On smoothness and generalized Lie theory}}\label{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 [[categorification]]s, as well as to their tangent structures like [[Lie algebroids]] and their interrelations ([[Lie theory]]: [[integration]], [[Lie integration]]).

\hypertarget{on_geometric_function_theory_and_quantization}{}\subsubsection*{{On geometric function theory and quantization}}\label{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]]\ldots{}).

\hypertarget{on_quantization_and_the_geometry_of_differential_operators}{}\subsubsection*{{On quantization and the geometry of differential operators}}\label{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 submanifold]]s 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 operator]]s. Sometimes it is possible or even useful to avoid fine analysis by using the [[algebraic approaches to differential calculus]] and [[regular differential operator|differential operators]], what also makes possible some noncommutative analogues.

\hypertarget{on_contructivism_and_computable_analysis}{}\subsubsection*{{On contructivism and computable analysis}}\label{on_contructivism_and_computable_analysis}

\begin{itemize}%
\item [[constructive analysis]], [[computable analysis]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[epsilontic analysis]]


\item [[infinitesimal analysis]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{ReferencesGeneral}{}\subsubsection*{{General}}\label{ReferencesGeneral}

Textbooks include

\begin{itemize}%
\item [[Walter Rudin]], \emph{Principles of Mathematical Analysis}, McGraw-Hill (1964, 1976) (\href{https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf}{pdf})


\item [[Eric Schechter]], \emph{[[Handbook of Analysis and its Foundations]]}, Academic Press (1996) (\href{http://www.math.vanderbilt.edu/~schectex/ccc/}{web})



\end{itemize}
Discussion of the history, amplifying its roots all the way back in [[Zeno's paradoxes of motion]] is in

\begin{itemize}%
\item Carl Benjamin Boyer, \emph{The history of the Calculus and its conceptual development}, Dover 1949

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Mathematical_analysis}{Mathematical analysis}}

\end{itemize}
See also the references at \emph{[[calculus]]}.

\hypertarget{ReferencesConstructiveAnalysis}{}\subsubsection*{{Constructive analysis}}\label{ReferencesConstructiveAnalysis}

The formulation of [[analysis]] in [[constructive mathematics]], hence [[constructive analysis]], was maybe inititated in

\begin{itemize}%
\item [[Errett Bishop]], \emph{Foundations of constructive analysis.} McGraw-Hill, (1967)

\end{itemize}
together with the basic notion of [[Bishop set]]/[[setoid]]. Implementations of constructive [[real number]] analysis in [[type theory]] implemented in [[Coq]] are discussed in

\begin{itemize}%
\item R. O'Connor, \emph{A Monadic, Functional Implementation of Real Numbers}. MSCS, 17(1):129-159, 2007 (\href{http://arxiv.org/abs/cs/0605058}{arXiv:0605058})


\item R. O'Connor, \emph{Certified exact transcendental real number computation in Coq}, In TPHOLs 2008, LNCS 5170, pages 246--261, 2008.


\item R. O'Connor, \emph{Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory}, PhD thesis, Radboud University Nijmegen, 2009.


\item Robbert Krebbers, [[Bas Spitters]], \emph{Type classes for efficient exact real arithmetic in Coq} (\href{http://arxiv.org/abs/1106.3448/}{arXiv:1106.3448})



\end{itemize}
[[!redirects mathematical analysis]]



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