Smooth Extensions in Infinite Dimensional Banach Spaces

Abstract

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 homeomorphism 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.

Extraction of critical points of smooth functions on Banach spaces

“…For this kind of sets the following abstract extractibility result holds. For more information about diffeomorphic extraction of closed sets in Banach spaces see for instance [8,24,20,21,10,1,2,3].…”

Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets

“…The proof of Theorem 1.4 combines ideas and techniques from Peter Renz's Ph.D. thesis [58], James West's paper [67], and some previous work of the first and second-named authors [4,5]; see Section 2 for more information. It should be noted that (even in the case that E is separable) Theorem 1.3 generalizes West's theorem [67], because a closed locally compact subset of an infinite-dimensional Hilbert space E, locally, can be regarded as the graph of a continuous mapping defined on a closed subset of an infinite-codimensional subspace of E; see, for instance, [58]. Furthermore, note in the above results we do not assume separability of the Banach space E.…”

“…In this section we will combine ideas and tools of [58,67,5] in order to prove Theorem 1.4. We will split the proof into four subsections.…”

“…Then, the composition ϕ −1 x • g x • h x will extract the local chunk of graph U x ∩ G(f x ) and will move no point too much. Finally, in Section 2.4, we will see how one can patch a collection of diffeomorphisms extracting pieces of X into a diffeomorphism h which extracts X and is limited by G. In Section 2.1, we will closely follow Peter Renz's results from [58,59]. In Sections 2.2 and 2.3, we will combine ideas and techniques from [58,4,5].…”

“…Finally, in Section 2.4, we will see how one can patch a collection of diffeomorphisms extracting pieces of X into a diffeomorphism h which extracts X and is limited by G. In Section 2.1, we will closely follow Peter Renz's results from [58,59]. In Sections 2.2 and 2.3, we will combine ideas and techniques from [58,4,5]. Finally, in Section 2.4, we will borrow a technique of James West's [67, p. 288-290].…”