Editing Examples of vector spaces (section)

==Polynomial vector spaces==
===One variable===

The set of [[polynomial]]s with coefficients in ''F'' is a vector space over ''F'', denoted ''F''[''x'']. Vector addition and scalar multiplication are defined in the obvious manner. If the [[Degree of a polynomial|degree of the polynomials]] is unrestricted then the dimension of ''F''[''x''] is [[countably infinite]]. If instead one restricts to polynomials with degree less than or equal to ''n'', then we have a vector space with dimension ''n''&thinsp;+&thinsp;1.

One possible basis for ''F''[''x''] is a [[monomial basis]]: the coordinates of a polynomial with respect to this basis are its [[coefficient]]s, and the map sending a polynomial to the sequence of its coefficients is a [[linear isomorphism]] from ''F''[''x''] to the infinite coordinate space ''F''<sup>∞</sup>.

The vector space of polynomials with real coefficients and degree less than or equal to ''n'' is often denoted by ''P''<sub>''n''</sub>.

===Several variables===
The set of [[polynomial]]s in several variables with coefficients in ''F'' is vector space over ''F'' denoted ''F''[''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''r''</sub>]. Here ''r'' is the number of variables.

{{See also|Polynomial ring}}