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===Generalized coordinate space===
Let ''X'' be an arbitrary set. Consider the space of all functions from ''X'' to ''F'' which vanish on all but a finite number of points in ''X''. This space is a vector subspace of ''F''<sup>''X''</sup>, the space of all possible functions from ''X'' to ''F''. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set.
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==Field extensions==
Suppose ''K'' is a [[Field extension|subfield]] of ''F'' (cf. [[field extension]]). Then ''F'' can be regarded as a vector space over ''K'' by restricting scalar multiplication to elements in ''K'' (vector addition is defined as normal). The dimension of this vector space, if it exists,{{efn|Note that the resulting vector space may not have a basis in the absence the [[axiom of choice]].}} is called the ''degree'' of the extension. For example the [[complex number]]s '''C''' form a two-dimensional vector space over the real numbers '''R'''. Likewise, the [[real numbers]] '''R''' form a vector space over the [[rational number]]s '''Q''' which has (uncountably) infinite dimension, if a Hamel basis exists.{{efn|There are models of [[
If ''V'' is a vector space over ''F'' it may also be regarded as vector space over ''K''. The dimensions are related by the formula
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