
Journal of Convex Analysis 19 (2012), No. 3, 685711 Copyright Heldermann Verlag 2012 MStructures in VectorValued Polynomial Spaces Verónica Dimant Dep. de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires  Argentina vero@udesa.edu.ar Silvia Lassalle Dep. de Matemática  Pab I, Fac. de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina slassall@dm.uba.ar [Abstractpdf] 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 vectorvalued polynomial context. Also, when $\mathcal P_w(^n E, F)$ is an $M$ideal in $\mathcal P(^n E, F)$ we prove a BishopPhelps type result for vectorvalued polynomials and relate normattaining polynomials with farthest points and remotal sets. Keywords: Mideals, homogeneous polynomials, weakly continuous polynomials on bounded sets MSC: 47H60,46B04,47L22,46B20 [ Fulltextpdf (245 KB)] for subscribers only. 