The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples...

Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {̂}_{n,\pi }E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\pi }E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {̂}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{\left[n\right]}$, the completion of the n-concavification of...

In this paper, the $ℱ$-Riesz norm for ordered $ℱ$-bimodules is introduced and characterized in terms of order theoretic and geometric concepts. Using this notion, $ℱ$-Riesz normed bimodules are introduced and characterized as the inductive limits of matricially Riesz normed spaces.

We deal with the weak Radon-Nikodým property in connection with the dual space of (X,Y), the space of compact operators from a Banach space X to a Banach space Y. First, under the weak Radon-Nikodým property, we give a representation of that dual. Next, using this representation, we provide some applications to the dual spaces of (X,Y) and ${}_{w*w}(X*,Y)$, the space of weak*-weakly continuous operators.

Several properties of weakly p-summable sequences and of the scale of p-converging operators (i.e., operators transforming weakly p-summable sequences into convergent sequences) in projective and natural tensor products with an lp space are considered. The last section studies the Dunford-Pettis property of order p (i.e., every weakly compact operator is p-convergent) in those spaces.