Editing Brouwer fixed-point theorem (section)

=== A proof using the hairy ball theorem ===
The [[hairy ball theorem]] states that on the unit sphere {{mvar|''S''}} in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field {{mvar|'''w'''}} on {{mvar|''S''}}. (The tangency condition means that {{mvar|'''w'''('''x''') ⋅ '''x'''}} = 0 for every unit vector {{mvar|'''x'''}}.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in {{harvtxt|Milnor|1978}}. 

In fact, suppose first that {{mvar|'''w'''}} is ''continuously differentiable''. By scaling, it can be assumed that {{mvar|'''w'''}} is a continuously differentiable unit tangent vector on {{mvar|'''S'''}}. It can be extended radially to a small spherical shell {{mvar|''A''}} of {{mvar|''S''}}. For  {{mvar|''t''}} sufficiently small, a routine computation shows that the mapping {{mvar|'''f'''<sub>''t''</sub>}}({{mvar|'''x'''}}) = {{mvar|'''x'''}} + {{mvar|''t'' '''w'''('''x''')}} is a [[contraction mapping]] on {{mvar|''A''}} and that the volume of its image is a polynomial in {{mvar|''t''}}. On the other hand, as a contraction mapping, {{mvar|'''f'''<sub>''t''</sub>}} must restrict to a homeomorphism of {{mvar|''S''}} onto (1 + {{mvar|''t''<sup>2</sup>}})<sup>{{sfrac|1|2}}</sup> {{mvar|''S''}} and  {{mvar|''A''}} onto (1 + {{mvar|''t''<sup>2</sup>}})<sup>{{sfrac|1|2}}</sup> {{mvar|''A''}}. This gives a contradiction, because, if the dimension {{mvar|''n''}} of the Euclidean space is odd, (1 + {{mvar|''t''<sup>2</sup>}})<sup>{{mvar|''n''}}/2</sup> is not a polynomial. 

If {{mvar|'''w'''}} is only a ''continuous'' unit tangent vector on {{mvar|''S''}}, by the [[Weierstrass approximation theorem]], it can be uniformly approximated by a polynomial map {{mvar|'''u'''}} of {{mvar|''A''}} into Euclidean space. The orthogonal projection on to the tangent space is given by {{mvar|'''v'''}}({{mvar|'''x'''}}) = {{mvar|'''u'''}}({{mvar|'''x'''}}) - {{mvar|'''u'''}}({{mvar|'''x'''}}) ⋅ {{mvar|'''x'''}}. Thus {{mvar|'''v'''}} is polynomial and nowhere vanishing on {{mvar|''A''}}; by construction {{mvar|'''v'''}}/||{{mvar|'''v'''}}|| is a smooth unit tangent vector field on {{mvar|''S''}}, a contradiction.    

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that {{mvar|''n''}} is even. If there were a fixed-point-free continuous self-mapping {{mvar|'''f'''}} of the closed unit ball {{mvar|''B''}} of the {{mvar|''n''}}-dimensional Euclidean space {{mvar|''V''}}, set

:<math>{\mathbf w}({\mathbf x}) =  (1 - {\mathbf x}\cdot {\mathbf f}({\mathbf x}))\, {\mathbf x} - (1 - {\mathbf x}\cdot {\mathbf x})\, {\mathbf f}({\mathbf x}).</math>

Since {{mvar|'''f'''}} has no fixed points, it follows that, for {{mvar|'''x'''}} in the [[interior (topology)|interior]] of {{mvar|''B''}}, the vector {{mvar|'''w'''}}({{mvar|'''x'''}}) is non-zero; and for {{mvar|'''x'''}} in {{mvar|''S''}},  the scalar  product <br/> {{mvar|'''x'''}} ⋅ {{mvar|'''w'''}}({{mvar|'''x'''}}) = 1 – {{mvar|'''x'''}} ⋅ {{mvar|'''f'''}}({{mvar|'''x'''}}) is strictly positive. From the original {{mvar|''n''}}-dimensional space Euclidean space {{mvar|''V''}}, construct a new auxiliary <br/>({{mvar|''n'' + 1}})-dimensional space {{mvar|''W''}} =  {{mvar|''V''}} x '''R''', with coordinates {{mvar|''y''}} = ({{mvar|'''x'''}}, {{mvar|''t''}}). Set

:<math>{\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).</math>

By construction {{mvar|'''X'''}} is a continuous vector field on the unit sphere of {{mvar|''W''}}, satisfying the tangency condition {{mvar|'''y'''}} ⋅ {{mvar|'''X'''}}({{mvar|'''y'''}})&nbsp;=&nbsp;0. Moreover, {{mvar|'''X'''}}({{mvar|'''y'''}}) is nowhere vanishing (because, if {{var|'''x'''}} has norm 1, then {{mvar|'''x'''}} ⋅ {{mvar|'''w'''}}({{mvar|''x''}}) is non-zero; while if {{mvar|'''x'''}} has norm strictly less than 1, then {{mvar|''t''}} and {{mvar|'''w'''}}({{mvar|'''x'''}}) are both non-zero). This contradiction proves the fixed point theorem when {{mvar|''n''}} is even. For {{mvar|''n''}} odd, one can apply the fixed point theorem to the closed unit ball {{mvar|''B''}} in {{mvar|''n'' + 1}} dimensions and the  mapping {{mvar|'''F'''}}({{mvar|'''x'''}},{{mvar|''y''}}) = ({{mvar|'''f'''}}({{mvar|'''x'''}}),0).
The advantage of this proof is that it uses only elementary techniques; more general results like the [[Borsuk-Ulam theorem]] require tools from [[algebraic topology]].<ref name="Milnor78">{{harvnb|Milnor|1978}}</ref>