A Banach space is polynomially Schur if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford–Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach ...

In [8] and [9] J. Jarník and J.Kurzweil introduced an integration process (called PU-integral) for real valued functions on an interval of n with the use of suitably regularC-partitions of unity, instead of the usual partitions. The PU-integral is nonabsolutely convergent and in dimension one falls properly in between the Lebesgue and the Kurzweil-Henstock integrals. In [4], without assuming an...

In 1953 A. Grothendieck introduced the property known as Dunford-Pettis property [18]. A Banach space X has the Dunford-Pettis property (DPP in the sequel) if whenever (xn) and (ρn) are weakly null sequences in X and X∗, respectively, we have ρn(xn) → 0. It is due to Grothendieck that every C(K )-space satisfies the DPP. Historically, were Dunford and Pettis who first proved that L1(μ) satisfie...

A type of oscillation modeled on BMO is introduced to characterize norm compactness in L 1. This result is used to characterize the bounded linear operators from L 1 into a Banach space X that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). This characterization is used to study the geometry of Banach spaces X with the property that all bounded linear o...

The paper studies the core-Walras equivalence problem in the commodity space framework of Banach spaces, allocations being defined as Pettis integrable functions. In particular, a core-Walras equivalence result for a certain class of commodity spaces is established, without requiring that the commodity space be separable. The class covered by this result includes the Lp(μ) spaces, 1 ≤ p < ∞, μ ...

Pettis raised the question whether or not separability of the range space implies almost everywhere weak differentiability of Pettis integrals. Phillips has given an example which answers this question in the negative. His construction is based on a sequence of orthogonal vectors in Hilbert space. We present here a different example of the same type of function. Our basic construction is that o...