Lipschitz functions on classical spaces

AbstractWe show that, for everyɛ > 0 and every Lipschitz functionf from the unit sphere of the Banach spacec0 to ℝ, there is an infinite-dimensional subspace ofc0, on the unit sphere of whichf varies by at most ɛ. This result is closely related to a theorem of Hindman, and a well known open problem in Banach space theory

Lipschitz functions on topometric spaces

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, normality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space $X$ from the lattice of continuous $1$-Lipschitz functions on $X$, in analogy with the recovery of a compact topological space $X$ from the structure of (real or complex) functions on $X$

Boundary-nonregular functions in the disc algebra and in holomorphic Lipschitz spaces

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Linear and Multilinear Isometries in a Noncompact Framework

Both classical linear and multilinear isometries defined between subalgebras of bounded continuous functions on (complete) metric spaces are studied. Particularly, we prove that certain such subalgebras, including the subalgebras of uniformly continuous, Lipschitz or locally Lipschitz functions, determine the topology of (complete) metric spaces. As consequence, it is proved that the subalgebra of Lipschitz functions determines the Lipschitz in the small structure of a complete metric space. Furthermore, we provide a weighted composition representation for multilinear isometries from similar subalgebras on (not necessarily complete) metric spaces. We apply this general representa- tion to obtain more specific ones for subalgebras of uniformly continuous and Lipschitz functions

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided $l(T) \times l(T^{-1})< 6/5$. By $l(T)$ and $l(T^{-1})$ we denote the Lipschitz constants of the maps $T$ and $T^{-1}$. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is 17/16. We also estimate the distance of the map $T$ from the isometry of the spaces $C(X)$ and $C(Y)$

Lip-density and algebras of Lipschitz functions on metric spaces.

Our aim in this note is to give an extension of the classical Myers-Nakai theorem in the context of Finsler manifolds. To achieve this, we provide a general result in this line for subalgebras of bounded Lipschitz functions on length metric spaces. We also establish some connection with the uniform approximation of bounded Lipschitz functions by functions in the subalgebra, keeping control on the Lipschitz constant