Quasi-differentiable functions of Banach spaces

Abstract

Abstract.Nonzero Fréchet differentiable functions with bounded support do not exist on certain real separable Banach spaces. As a result, the class of differentiable functions on such spaces is too small to be useful. For example, the class of differentiable functions on certain spaces does not separate disjoint closed subsets of the space. It is shown that this separation problem does not arise if Fréchet differentiability is replaced by the weaker condition of quasi-differentiability.Furthermore, it is shown that any bounded uniformly continuous function on a real separable Banach space is the uniform limit of quasi-differentiable functions.

Partitions of Unity in Sobolev Spaces over Infinite Dimensional State Spaces

“…In the case of the Sobolev spaces D r p over an abstract Wiener space we even prove the existence of partitions of unity which are simultaneously in all of them, i.e., in the Malliavin test function space D . This was, however, essentially known (see [Go71], [Pi77] and Remark 2.2(ii)), but our general techniques developed in Section 1 below, lead to a quite simple new proof of this fact, which is presented in Section 2. Subsequently, we concentrate on partitions of unity in Sobolev type spaces of order one, more precisely Dirichlet spaces.…”

“…[Go71] [Pi77]). But it was surprising for us that nobody realized that the functions used in [Go71] can be used to construct partitions of unity of class W (with W as in Theorem 2.1).…”

“…[Go71] [Pi77]). But it was surprising for us that nobody realized that the functions used in [Go71] can be used to construct partitions of unity of class W (with W as in Theorem 2.1). For example, F. Getzler erroneously stated in [Ge86] that``Unfortunately, since Wiener spaces are not locally compact, D partitions of unity do not exist in general.…”

Gaussian measures on linear spaces

“…Nice differentiabiLity properties of Gaussian measures can be used for constructing partitions of unity and smooth approximations consisting of functions infinitely differentiable along a dense subspace H. Some results in this directions are contained in [33,43,205,216,366].…”

On Gross differentiation on Banach spaces

“…The works [1; 7], among others, show that for many separable Banach spaces the bounded continuously Frechet diίferentiable functions are not dense in the space of bounded uniformly continuous functions. However, by regarding a real separable Banach space B as an abstract Wiener space [3], Goodman [2] is able to show that the set of bounded continuously quasi-diίferen-tiable functions on B is dense in the space of bounded uniformly continuous functions on B. Regarding B as an abstract Wiener space has a more important advantage, namely, we can talk about whether the second derivative is a Hilbert-Schmidt or trace class operator.…”