In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an $n$-normed space, we are interested in bounded multilinear $n$-functionals and $n$-dual spaces. The concept of bounded multilinear $n$-functionals on an $n$-normed space was initially intoduced by White (1969), and studied further by Batkunde et al., and Gozali et al. (2010). In this paper, we revisit the definition of bounded multilinear $n$-functionals, introduce the concept of $n$-dual spaces, and...

We study the numerical radius of Lipschitz operators on Banach spaces. We give its basic properties. Our main result is a characterization of finite-dimensional real Banach spaces with Lipschitz numerical index 1. We also explicitly compute the Lipschitz numerical index of some classical Banach spaces.

We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.

Let $G$ be a locally compact group, and let $\parallel {\parallel }_{\infty }^B$ be a function norm on $L^1(G{)}_{\mathrm{loc}}$ such that the space $L^{\infty }(G,B)$ of all locally integrable functions with finite $\parallel {\parallel }_{\infty }^B$-norm is an invariant solid Banach function space. Consider the space $L_{\mathrm{RUC}}(G,B)$ of all functions in $L^{\infty }(G,B)$ of which the right translation is a continuous map from $G$ into $L^{\infty }(G,B)$. Characterizations of the case where $L_{\mathrm{RUC}}(G,B)$ is a Riesz ideal of $L^{\infty }(G,B)$ are given in terms of the order-continuity of $\parallel {\parallel }_{\infty }^B$ on certain subspaces of $L^{\infty }\left(G\right)$. Throughout the paper, the discussion is carried out in the context...

We establish necessary and sufficient conditions under which the linear span of positive AM-compact operators (in the sense of Fremlin) from a Banach lattice $E$ into a Banach lattice $F$ is an order $\sigma$-complete vector lattice.

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces...

Let $X$ be a Banach lattice, and denote by $X_{+}$ its positive cone. The weak topology on $X_{+}$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1\left(\Gamma \right)$ for a set $\Gamma$. The weak${}^{*}$ topology on $X_{+}^{*}$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C\left(K\right)$-space, where $K$ is a metrizable compact space.

Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_0$. Then the Bochner space $L^1(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.

In this paper we discuss the problem of when the projective tensor product of two Banach spaces has the Radon-Nikodym property. We give a detailed exposition of the famous examples of Jean Bourgain and Gilles Pisier showing that there are Banach spaces X and Y such that each has the Radon-Nikodym property but for which their projective tensor product does not; this result depends on the classical theory of absolutely summing, integral and nuclear operators, as well as the famous Grothendieck inequality...

A characterization of property $\left(\beta \right)$ of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space $l_{\Phi }\left(X\right)$ has the property $\left(\beta \right)$ if and only if both spaces $l_{\Phi }$ and $X$ have it also. In particular the Lebesgue-Bochner sequence space $l_p\left(X\right)$ has the property $\left(\beta \right)$ iff $X$ has the property $\left(\beta \right)$. As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property $\left(\beta \right)$, nearly uniform convexity, the drop property and...

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of...