A NONCOMPACT DIRICHLET NORM

Abstract

has finite dimensional homology at C (M,B). In this note we show that the inclusion B -L2(M,B) may fail to be compact even for rather simple elliptic complexes. We shall use the following lemma. LEMMA. Assume that the inclusion B -, L2(M,B) is compact. Let {}uj be a sequence in C (M,A) such that uj I ker S and 11 )uj1 1. Then a subsequence of {ujj converges in the L2 norm. Proof: According to Theorem 3 of reference 1, there exists a sequence {vj} in N such that (5DD* + 8*8)vj = a)uj and llvjl[ < c, where c is a constant independent of j. Since l*vjl2 + 18vj12 + j+ V j2 = ({u,,vj) + Ilv, 2 ( + C2, the compactness assumption implies that a subsequence of vj} converges in the L2 norm. Now in the equation (5S)* + 8*8)vj = S)uj, the term 8*8v is orthogonal to the other two terms; hence OO)D*vj = Duj and 8*8vj = 0. Also, since both D*vj and uj are orthogonal to ker S, we must have uj = DO*vj. The inequality