Geometry of the Heat Equation

where jX2 must converge. And, in this case, G also will be a Hilbert space. We use a very valuable theorem due to Banach4 to show that if GI denotes a separable metric space of type' B, and if the fundamental operation is that of addition, then we can find a denumerable set of characters of G' such that, at any rate formally, every continuous-character can be expressed as a finite or infinite product of powers of these fundamental characters. 1 INTERNATIONAL RESEARCH FELLOW. 2 A. Haar, "Uber unendliche komutative Gruppen," Math. Zeit., 33, 129-159 (1931). 3F. Peter and H. Weyl, "Die Vollstiandigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe," Math. Ann., 97, 737-755 (1927). 4 S. Banach, Theorie des op.rations lin&&ires, Warszawa, 1932. 5See, e.g., Banach, loc. cit., Chapter V.