Editing Normed vector space (section)

==Linear maps and dual spaces==

The most important maps between two normed vector spaces are the [[Continuous function (topology)|continuous]] [[Linear transformation|linear maps]]. Together with these maps, normed vector spaces form a [[Category theory|category]].

The norm is a continuous function on its vector space.  All linear maps between finite-dimensional vector spaces are also continuous.

An ''isometry'' between two normed vector spaces is a linear map <math>f</math> which preserves the norm (meaning <math>\|f(\mathbf{v})\| = \|\mathbf{v}\|</math> for all vectors <math>\mathbf{v}</math>). Isometries are always continuous and [[injective]]. A [[surjective]] isometry between the normed vector spaces <math>V</math> and <math>W</math> is called an ''isometric isomorphism'', and <math>V</math> and <math>W</math> are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes.

When speaking of normed vector spaces, we augment the notion of [[dual space]] to take the norm into account. The dual <math>V^{\prime}</math> of a normed vector space <math>V</math> is the space of all ''continuous'' linear maps from <math>V</math> to the base field (the complexes or the reals) — such linear maps are called "functionals".  The norm of a functional <math>\varphi</math> is defined as the [[supremum]] of <math>|\varphi(\mathbf{v})|</math> where <math>\mathbf{v}</math> ranges over all unit vectors (that is, vectors of norm <math>1</math>) in <math>V.</math> This turns <math>V^{\prime}</math> into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the [[Hahn–Banach theorem]].