New concepts related to approximating a Lipschitz function between Banach spaces by a ne functions are introduced Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X Y for which any Lipschitz function from X to Y can be so approx imated is obtained This is applied to the study of Lipschitz and uniform quotient mapp...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...

Jachymski [ Proc. Amer. Math. Soc., 136 (2008), 1359-1373] gave modified version of a Banach fixed point theorem on a metric space endowed with a graph. In the present paper, (G, Φ)-graphic contractions have been de ned by using a comparison function and studied the existence of fixed points. Also, Hardy-Rogers G-contraction have been introduced and some fixed point theorems have been proved. S...

We prove two results about smoothness in Banach spaces of the type C(K). Both build upon earlier papers of the first named author [3,4]. We first establish a special case of a conjecture that remains open for general Banach spaces and concerns smooth approximation. We recall that a bump function on a Banach space X is a function β : X → R which is not identically zero, but which vanishes outsid...

In an earlier paper [4], the author sketched a method, based on the use of “Talagrand operators”, for defining infinitely differentiable equivalent norms on the spaces C0(L) for certain locally compact, scattered spaces L. A special case of this result was that a C renorming exists on C0(L) for every countable locally compact L. Recently, Hájek [3] extended this result by showing that a real no...

Let H be a separable Banach space. We considered the sequence of stochastic integrals {Xn− · Yn} where {Yn} is a sequence of infinite dimesnional H semimartingales and Xn are H valued cadlag processes. Assuming that {(Xn, Yn)} satisfies large deviation principle, a uniform exponential tightness condition is described under which large deviation principle holds for {(Xn, Yn, Xn− · Yn)}. When H i...

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX : X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions f |B◦ X to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersecti...

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to ...

Assuming the continuum hypothesis, we construct a universally weakly measurable function from [0,1] into a dual of some weakly compactly generated Banach space, which is not Pettis integrable. This (partially) solves a problem posed by Riddle, Saab and Uhl [13]. We prove two results related to Pettis integration in dual Banach spaces. We also contribute to the problem whether it is consistent t...