Editing Group ring (section)

===Interpretation as functions===
Thinking of the [[free vector space]] as ''K''-valued functions on ''G'', the algebra multiplication is [[convolution]] of functions.

While the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for [[cofinitely]] many points; topologically (using the [[discrete topology]]), these correspond to functions with [[compact support]].

However, the group algebra ''K''[''G''] and the space of functions {{nowrap|1=''K''<sup>''G''</sup> := Hom(''G'', ''K'')}} are dual: given an element of the group algebra

:<math>x = \sum_{g\in G} a_g g</math>

and a function on the group {{nowrap|''f'' : ''G'' → ''K''}} these pair to give an element of ''K'' via

:<math>(x,f) = \sum_{g\in G} a_g f(g),</math>

which is a well-defined sum because it is finite.