transactions of the american mathematical society Volume 168, June 1972 SMOOTH EXTENSIONS IN INFINITE DIMENSIONAL BANACH SPACES BY PETER RENZf1) Abstract. If B is /„(a>) or c0(<u)we show B has the following extension property. Any homeomorphism from a compact subset M of B into B may be extended to a homeomorphism of B onto B which is a C" diffeomorphism on B\M to its image in B. This is done by writing B as a direct sum of closed subspaces B, and jB2 both isomorphically isometric to B so that the natural projection of K into B, along B2 is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With K, B, B, and B2 as above a homeo- morphism of B onto itself is constructed which leaves the Si-coordinates of points in B unchanged, carries K into B, and is a C" diffeomorphic map on B\K. From these results the extension theorem may be proved by standard methods. 1. Introduction, outline and preliminaries. Our main result is the following extension theorem. Theorem 4.1. Let B be C0(<7) or lp(w) and let M and N be compact subsets of B. Let f be a homeomorphism mapping M onto N. There is an extension f* of f to a homeomorphism of B onto itself such that f* restricted to B\M isa C œ diffeomorphism of B\M onto B\N. We start with the following Splitting Theorem which is an immediate consequence of the results and techniques of Corson [6]. Splitting Theorem (Corson). Let B be C0(oj) or Ip(oj). Let K be a o-compact subset of B. Then B may be written as an internal direct sum B=BX + B2 where Bx and B2 are both isomorphic to B and where the projection -nx of B into Bx along B2 yields a one-to-one map when restricted to K. This theorem will be applied to a compact set K and -rrx(K) will be denoted Kx. In this case 77! yields a homeomorphism of K onto Kx. Further, there is a con- tinuous mapping g of Kx into Bx such that K={kx+g(kx) \ kx e Kx}. By a slight abuse of the language we call {kx+g(kx) \ kx e Kx} the graph of g. Thus we see that the compact set K is the graph of a continuous function g from some set Kx which lies in a subspace Bx of infinite deficiency in B into B2 a complement of Bx in B. Presented to the Society, April 26, 1969 under the title Smooth extensions and smooth extractions; received by the editors January 7, 1971 and, in revised form, April 26, 1971. AMS 1970 subject classifications. Primary 57D50, 54C20; Secondary 58B10. Key words and phrases. Extension of homeomorphisms, infinite dimensional Banach spaces, C" extensions, compact subsets of infinite dimensional spaces. I1) The preparation of this paper was aided by support from Reed College. Copyright © 1972, American Mathematical Society 121