The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

In recent years, the diamondback moth, Plutella xylostella (Lepidoptera: Yponomeutidae), has become the most destructive insect of cruciferous plants throughout the world, and the annual cost for managing it is estimated to be US $I billion (168) Members of the plant family Cruciferae occur temperate and tropical climates and represent a diverse, widespread, and important plant group that includes cabbage, broccoli, cauliflower, collards, rapeseed, mustard, and Chinese cabbage, the most important vegetable crop grown in China (90), the most populous country in the world Although the diamondback moth is believed to have originated in the Mediterranean area (64), the source of some of our most important crucifers (185), diamondback moths now occur wherever cmcifers are grown, and this insect is believed to be the most universally distributed of all Lepidoptera (107) Absence of effective natural enemies, especially parasitoids, is believed to be a major cause of the diamondback moth’s pest status in most parts of the world (92) Lack of parasitoids in a particular area may have occurred because the diamondback moth is better able than its natural-enemy complex to become established in newly planted cmcifers Reports on the ability of diamondback moths to migrate long distances are numerous (19, 40, 54, 58, 108, 120,

http://web.entomology.cornell.edu/shelton/publications/pdf/Talekar-Shelton_1993_Ann_Rev_Ent_DBM.pdf

Biology, Ecology, and Management of the Diamondback Moth

The survey is devoted to the modern state of the theory of interpolation of linear operators acting in Banach spaces. Principal attention is devoted to real and complex methods and applications of the theory of interpolation to analysis.

Interpolation of Linear Operators

Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C ¤ -algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming’ and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes.

/pdf/grothendieck-s-theorem-past-and-present-1wekbrrlwf.pdf

Grothendieck's Theorem, past and present

Homological properties of associative algebras arising in the theory of helices are studied. A class of noncommutative algebras is introduced in which it is natural (from the viewpoint of the theory of helices) to deform projective spaces and also certain Fano varieties. It is shown that in the case of deformations of the projective plane this approach leads to algebras associated with automorphisms of two-dimensional cubic curves.

https://iopscience.iop.org/article/10.1070/IM1994v042n02ABEH001536/pdf

Homological properties of associative algebras: the method of helices

/pdf/projective-tensor-products-of-c-algebras-1yruwnrp9g.pdf

Projective tensor products of $C^*$-algebras.

Hardy spaces for the strip

Preliminaries.- The real method.- The complex method.- Categorical notions.- Finite dimensional doolittle diagrams.- Kan extensions.- Duality.- More about duality.- The classical methods from a categorical viewpoint.

Interpolation Functors and Duality

Real interpolation of Banach algebras and factorization of weakly compact homomorphisms

We define the $k$:th moment of a Banach space valued random variable as the expectation of its $k$:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. 
We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. 
One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. 
Several sections are devoted to results in special Banach spaces, including Hilbert spaces, $C(K)$ and $D[0,1]$. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in $D[0,1]$ that we need. 
One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.

Higher moments of Banach space valued random variables

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