The book under review, Topological Vector Spaces, Distributions, and Kernels, by François Trèves, is a 2006 Dover Publications re-issue of the well-known book, by the same title, originally published by Academic Press in 1967. Since the familiar green hardcover Academic Press books are pretty hard to find nowadays, be it in second-hand bookstores or via on-line second-hand booksellers, a Dover offering is particularly welcome, especially in light of its characteristic affordability. Trèves’ book is a wonderful choice, given its established importance and continuing relevance.
As Trèves points out in his Preface, now over forty years old but still on the mark, the goal of Topological Vector Spaces, Distributions, and Kernels is to “[acquaint] the graduate student with that section of functional analysis which reaches beyond the boundaries of Hilbert spaces and Banach spaces theory and whose influence is now deeply felt in analysis, particular in the field of partial differential equations.” Thus, the reader had better be specifically oriented toward the indicated part of “hard analysis” and should perhaps already have some solid functional analysis under his belt.
Indeed, while Trèves properly informs the reader that his exposition generally starts with basics, e.g. the definitions of a vector space and a topological space (launching Part I of the book), “the difficulty of the reading increases.” To wit, Part I, devoted to topological vector spaces and spaces of functions, also introduces the eclectic notion of filter very early on and, in pretty short order, reaches Cauchy filters and their application to the important concept of Cauchy completion. And this 173-page long first part also covers Fréchet spaces, for instance the special case of spaces of infinitely differentiable functions rapidly decreasing at infinity, critical to so many areas of modern hard analysis, PDE in particular. We’re flying at 30,000 feet in no time at all…
Part I of Topological Vector Spaces, Distributions, and Kernels ends with a truly superb seven-page (!) treatment of the open mapping theorem, capped off by four hugely important corollaries including half of the closed graph theorem (for metrizable and complete topological vector spaces). This illustrates the density of the book, of course, and again underscores the fact that Topological Vector Spaces, Distributions, and Kernels is meant for the serious, committed, and more or less mature reader, a fledgling analyst already a few years along in his specialty.
Part II of the book deals with the important subjects of duality and distributions without which contemporary analysis is almost unthinkable. Trèves’ discussion of the latter subject is especially noteworthy, with §21, titled “ Radon measures and distributions,” a true gem, both for its compact elegance (at only ten pages: Trèves doesn’t waste words) and for its depth. Then Part III, “Tensor products. Kernels,” can almost be regarded as a course in itself on multilinearity in the context of completed topological spaces. Says Trèves: “…Parts I and II correspond, more or less, to a one year course… Part III has been added because I firmly believe that analysts should have some familiarity with tensor products, their natural topologies, and their completions…” Voilà.
Finally, Topological Vector Spaces, Distributions, and Kernels comes equipped with a decent number of exercises which are, I think, somewhat reminiscent of those found in another famous entry in the same Academic Press series, Jean Dieudonné’s Foundations of Modern Analysis, which is to say that the exercises are difficult but not impossible, important but not critical to the flow of the narrative, and will take the reader, or student, from adequate to considerably deeper understanding and facility. Topological Vector Spaces, Distributions, and Kernels is certainly worthy of its reputation.
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles.