Restricted product

The restricted product is a construction in the theory of topological groups.

Let <math> (G_1, K_1), (G_2, K_2) ,(G_3,K_3), ... </math> be sequence of locally compact groups together with a compact subgroup <math> K_i \subset G_i</math>. The restricted product

<math> \prod_i' G_i </math>

is the set of sequences <math> (g_1, g_2, ...) </math> such that <math> g_i \in K_i </math> for all but finitely many <math> i</math>. The restricted product is itself a locally compact Group. The best known example of this construction is that of the adele ring of an algebraic number field.