Editing Vector space (section)

==History==

Vector spaces stem from [[affine geometry]], via the introduction of [[coordinate]]s in the plane or three-dimensional space. Around 1636, French mathematicians [[René Descartes]] and [[Pierre de Fermat]] founded [[analytic geometry]] by identifying solutions to an equation of two variables with points on a plane [[curve]].{{sfn|Bourbaki|1969|loc = ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91}} To achieve geometric solutions without using coordinates, [[Bernhard Bolzano|Bolzano]] introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.{{sfn|Bolzano|1804}} {{harvtxt|Möbius|1827}} introduced the notion of [[Barycentric coordinates (mathematics)|barycentric coordinates]].{{sfn|Möbius|1827}} {{harvtxt|Bellavitis|1833}} introduced an [[equivalence relation]] on directed line segments that share the same length and direction which he called [[equipollence (geometry)|equipollence]].{{sfn|Bellavitis|1833}} A [[Euclidean vector]] is then an [[equivalence class]] of that relation.{{sfn|Dorier|1995}} 

Vectors were reconsidered with the presentation of [[complex numbers]] by [[Jean-Robert Argand|Argand]] and [[William Rowan Hamilton|Hamilton]] and the inception of [[quaternion]]s by the latter.{{sfn|Hamilton|1853}} They are elements in '''R'''<sup>2</sup> and '''R'''<sup>4</sup>; treating them using [[linear combination]]s goes back to [[Laguerre]] in 1867, who also defined [[system of linear equations|systems of linear equations]].

In 1857, [[Arthur Cayley|Cayley]] introduced the [[matrix notation]] which allows for harmonization and simplification of [[linear map]]s. Around the same time, [[Grassmann]] studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.{{sfn|Grassmann|2000}} In his work, the concepts of [[linear independence]] and [[dimension]], as well as [[scalar product]]s are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called [[Algebras over a field|algebras]]. Italian mathematician [[Giuseppe Peano|Peano]] was the first to give the modern definition of vector spaces and linear maps in 1888,{{sfn|Peano|1888|loc = ch. IX}} although he called them "linear systems".{{sfn|Guo|2021}} Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897, [[Salvatore Pincherle]] adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.{{sfn|Moore|1995|pages=268-271}} 

An important development of vector spaces is due to the construction of [[function spaces]] by [[Henri Lebesgue]]. This was later formalized by [[Stefan Banach|Banach]] and [[David Hilbert|Hilbert]], around 1920.{{sfn|Banach|1922}} At that time, [[algebra]] and the new field of [[functional analysis]] began to interact, notably with key concepts such as [[Lp space|spaces of ''p''-integrable functions]] and [[Hilbert space]]s.{{sfnm
 | 1a1 = Dorier | 1y = 1995
 | 2a1 = Moore | 2y = 1995
}}