<sod>Vector spaces are the subject of [[linear algebra]] and are well characterized by their [[dimension (vector space)|dimension]], which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in [[mathematical analysis]], as [[function space]]s, whose vectors are [[function (mathematics)|functions]]. These vector spaces are generally endowed with additional structure, which may be a [[topology]], allowing the consideration of issues of proximity and [[continuous function|continuity]]. Among these topologies, those that are defined by a [[norm (mathematics)|norm]] or [[inner product]] are more commonly used, as having a notion of [[metric (mathematics)|distance]] between two vectors. This is particularly the case of [[Banach space]]s and [[Hilbert space]]s, which are fundamental in mathematical analysis.<eod>
<sod>Vector spaces are the subject of [[linear algebra]] and are well-characterized by their [[dimension (vector space)|dimension]], which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in [[mathematical analysis]], as [[function space]]s, whose vectors are [[function (mathematics)|functions]]. These vector spaces are generally endowed with additional structure, which may be a [[topology]], allowing the consideration of issues of proximity and [[continuous function|continuity]]. Among these topologies, those that are defined by a [[norm (mathematics)|norm]] or [[inner product]] are more commonly used, as having a notion of [[metric (mathematics)|distance]] between two vectors. This is particularly the case of [[Banach space]]s and [[Hilbert space]]s, which are fundamental in mathematical analysis.<eod>