Remotality of certain sets Lp(I,X)

Let X be a Banach space and let (I, Ω, µ) be a measure space. For 1 ≤ p < ∞, let Lp (I, X) denote the space of Bochner p−integrable functions defined on I with values in X. The object of this paper is to give sufficient conditions for remotality of L1 (I, H) + L1 (I, G) in L1 (I, X), where H and G are two bounded sets in X which include as a special case remotality of L1 (I) ∧ ⊗ G + H ∧ ⊗ L1 (I) in L1 (I × I).Publisher's Versio

MM-structures in vector-valued polynomial spaces

This paper is concerned with the study of $M$-structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal P_w(^n E, F)$, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal P(^n E, F)$. We show that there is some hope for this to happen only for a finite range of values of $n$. We establish sufficient conditions under which the problem has positive and negative answers and use the obtained results to study the particular cases when $E=\ell_p$ and $F=\ell_q$ or $F$ is a Lorentz sequence space $d(w,q)$. We extend to our setting the notion of property $(M)$ introduced by Kalton which allows us to lift $M$-structures from the linear to the vector-valued polynomial context. Also, when $\mathcal P_w(^n E, F)$ is an $M$-ideal in $\mathcal P(^n E, F)$ we prove a Bishop-Phelps type result for vector-valued polynomials and relate norm-attaining polynomials with farthest points and remotal sets