Differential Invariants and Invariant Partial Differential Equations Under Continuous Transformation Groups in Normed Linear Spaces.

H1 c .-lp'(U n H1) = Hi_1(U n Hi) c Hi-,U.Hrl/Hi-,isconnected,so we can choose hi so that p(h1) is a boundary point of so(U n Hi). If hi e Hi-1 V then hiV c Hi-1 U and, relative to H,, hi would be an interior point of the sets hiV n Hi c Hi_1U n H, = Hi_1(U n Hi); whence, o being an open mapping, .p(hi) would be interior to jo(Hi-(U n Hi)) = 'o(U n Hi). Thus, we have obtained an infinite sequence of elements of U which satisfy the relation hi-lhj non-e Vfor i < j. This contradicts the compactness of U. LEMMA 2. If {Ha } is an increasing collection oj connected compact subgroups of a locally compact group G, then ( u Ha)is a connected compact subgroup. Proof: That it is a connected subgroup is trivial. Choose a compact neighborhood U of e in G, and an index 8 as in Lemma 1. Then (u Ha) is compact being a closed subset of H, U. The theorem now follows directly from Zorn's Lemma. In the proof we have used only the connectedness of certain factor spaces; hence we can strengthen our result slightly: COROLLARY. Every compact subgroup H of a locally compact group is contained in a compact subgroup which is maximal among those compact subgroups K for which K/H is connected.