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===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
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>.
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