Johan Swart

Banach ideals of p-compact operators

AbstractThe main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through lp (1≤p<∞) or co, called p-compact and ∞-compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p=∞ they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete ɛ-tensor product

Addendum to the paper The Projective Tensor Product II: The Radon-Nikodym Property

\ell^{P}\tilde{\bigotimes}_{\varepsilon}E

The role of ()-spaces and ¹-spaces

(1≤p<∞) is given.

norms related to Hilbert space

This chapter discusses about the Riesz theorem. The use of the Riesz theorem that is used in tandem with the Hahn–Banach theorem is examined in the chapter. One of the first applications of the Riesz theorem is in characterizing weakly convergent sequences in metric spaces. The partnership of the Riesz Theorem with various forms of the Hahn–Banach theorem is the most powerful in all abstract analysis. The most basic form of the Hahn–Banach theorem extending bounded linear functional from a subspace of a Banach space to the whole space without changing the norm is analyzed in the chapter. The Banachs version of the extension theorem is applied wherein a linear functional that is dominated by a positively homogeneous, subadditive functional is extended to the whole space with domination remaining. One remarkable consequence of Choquets theorem is the characterization of weakly null sequences in Banach spaces. It is shown that between Hilbert spaces, the 2-summing operators and Hilbert–Schmidt operators are precisely the same, while the trace-class coincides with the nuclear operators.

The problems of the Résumé

“(BY ∗∗ , weak). To see that (BX∗∗ , weak∗) × BY is υ-measurable, we call on Choquet’s theory of capacities and K-analytic sets. More specifically, since BY is Polish it is a Kσδ set; it follows that (BX∗∗ , weak ∗) × BY is a Kσδ subset of K , its Cech-Stone compactification. As such it is universally “f -capacitable”, where f ranges over all the regular capacities defined on K; in particular, it is universally measurable with respect to the regular Borel measures on K [cf. G. Choquet, Lectures on Analysis, Vol. I, W. A. Benjamin, Inc., 1969, especially pp. 141–156]. Moreover