Editing Vector space (section)

===Vector bundles===
{{Main|Vector bundle|Tangent bundle}}
[[Image:Mobius strip illus.svg|class=skin-invert-image|thumb|249px|right|A Möbius strip. Locally, it [[homeomorphism|looks like]] {{math|''U'' × '''R'''}}.]]
A ''vector bundle'' is a family of vector spaces parametrized continuously by a [[topological space]] ''X''.{{sfn|Spivak|1999|loc = ch. 3}} More precisely, a vector bundle over ''X'' is a topological space ''E'' equipped with a continuous map 
<math display=block>\pi : E \to X</math>
such that for every ''x'' in ''X'', the [[fiber (mathematics)|fiber]] π<sup>−1</sup>(''x'') is a vector space. The case dim {{math|1=''V'' = 1}} is called a [[line bundle]]. For any vector space ''V'', the projection {{math|''X'' × ''V'' → ''X''}} makes the product {{math|''X'' × ''V''}} into a [[trivial bundle|"trivial" vector bundle]]. Vector bundles over ''X'' are required to be [[locally]] a product of ''X'' and some (fixed) vector space ''V'': for every ''x'' in ''X'', there is a [[neighborhood (topology)|neighborhood]] ''U'' of ''x'' such that the restriction of π to π<sup>−1</sup>(''U'') is isomorphic<ref group=nb>That is, there is a [[homeomorphism]] from π<sup>−1</sup>(''U'') to {{math|''V'' × ''U''}} which restricts to linear isomorphisms between fibers.</ref> to the trivial bundle {{math|''U'' × ''V'' → ''U''}}. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space ''X'') be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle {{math|''X'' × ''V''}}). For example, the [[Möbius strip]] can be seen as a line bundle over the circle ''S''<sup>1</sup> (by [[homeomorphism#Examples|identifying open intervals with the real line]]). It is, however, different from the [[cylinder (geometry)|cylinder]] {{math|''S''<sup>1</sup> × '''R'''}}, because the latter is [[orientable manifold|orientable]] whereas the former is not.{{sfn|Kreyszig|1991|loc=§34, p. 108}}

Properties of certain vector bundles provide information about the underlying topological space. For example, the [[tangent bundle]] consists of the collection of [[tangent space]]s parametrized by the points of a differentiable manifold. The tangent bundle of the circle ''S''<sup>1</sup> is globally isomorphic to {{math|''S''<sup>1</sup> × '''R'''}}, since there is a global nonzero [[vector field]] on ''S''<sup>1</sup>.<ref group=nb>A line bundle, such as the tangent bundle of ''S''<sup>1</sup> is trivial if and only if there is a [[section (fiber bundle)|section]] that vanishes nowhere, see {{harvtxt|Husemoller|1994}}, Corollary 8.3. The sections of the tangent bundle are just [[vector field]]s.</ref> In contrast, by the [[hairy ball theorem]], there is no (tangent) vector field on the [[2-sphere]] ''S''<sup>2</sup> which is everywhere nonzero.{{sfn|Eisenberg|Guy|1979}} [[K-theory]] studies the isomorphism classes of all vector bundles over some topological space.{{sfn|Atiyah|1989}} In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real [[division algebra]]s: '''R''', '''C''', the [[quaternion]]s '''H''' and the [[octonion]]s '''O'''.

The [[cotangent bundle]] of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the [[cotangent space]]. [[Section (fiber bundle)|Sections]] of that bundle are known as [[differential form|differential one-form]]s.