What do you think?
Rate this book
Topological Vector Spaces, Distributions and Kernels
This text for upper-level undergraduates and graduate students focuses on key notions and results in functional analysis. Extending beyond the boundaries of Hilbert and Banach space theory, it explores aspects of analysis relevant to the solution of partial differential equations. It features basic classical results, plus 390 exercises. 1967 edition.
- GenresMathematics
565 pages, Hardcover
First published January 1, 1967
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
June 12, 2017
I'm torn in reviewing Trèves. This book is a modern classic, and you'll struggle to get very far in many areas that depend heavily on functional analysis without seeing references to it. While the prose are wonderful and there are some advantages the book's presentation, I find Trèves as challenging to use as a reference now as I found it to read when learning subject. I think this largely stems from Trèves's focus on classical function spaces on Euclidean domains and, in conjunction with this, an absence of category theory. While classical function spaces are all well and good, I think texts like Schaefer's Topological Vector Spaces do a much better job presenting concepts in a reasonably abstract, categorical way and then restricting to particular examples for the sake of illustration.
A great example of this is Trèves's presentation of (strict) LF-spaces, which are limits of countable ascending chains of Fréchet spaces, in which the topology of the n-th space properly contained in the (n+1)-st and its topology is also that induced on it by the (n+1)-st space. These spaces are obviously inductive limits of Fréchet spaces, so a modern student is wont to ask if the inductive limit is defined on the category of locally convex spaces, Hausdorff topological vector spaces, etc., and also what the properties of this limit are more generally. Trèves introduces these spaces and studies them without ever discussing why he is assuming the chain is countable, why he assumes it is a chain (as opposed to a directed family), what happens when you want to try this with more specialized or general topological vector spaces, and so on. This is in stark contrast to Schaefer, who begins with inductive topologies on arbitrary directed families of topological vector spaces and proceeds to specialize to a number of special cases. This is either before or after he also provides a treatment of projective limits of these spaces. In each case the generality makes the context clearer and allows you to contextualize what these objects are in the grand categorical scheme of things. (...no "scheme" puns intended...)
I also would say his treatment of nuclear spaces has become outdated, even when compared to a relatively old text like the excellent survey Nuclear Locally Convex Spaces by Pietsch. At any rate, your mileage may vary, and it's far too ubiquitous a classic not to familiarize yourself with. And if you're willing to put in the time, the exercises are excellent, albeit again overly specialized and not exploring the edge cases and exceptions that I think really help flesh out one's understanding of a theory.
A great example of this is Trèves's presentation of (strict) LF-spaces, which are limits of countable ascending chains of Fréchet spaces, in which the topology of the n-th space properly contained in the (n+1)-st and its topology is also that induced on it by the (n+1)-st space. These spaces are obviously inductive limits of Fréchet spaces, so a modern student is wont to ask if the inductive limit is defined on the category of locally convex spaces, Hausdorff topological vector spaces, etc., and also what the properties of this limit are more generally. Trèves introduces these spaces and studies them without ever discussing why he is assuming the chain is countable, why he assumes it is a chain (as opposed to a directed family), what happens when you want to try this with more specialized or general topological vector spaces, and so on. This is in stark contrast to Schaefer, who begins with inductive topologies on arbitrary directed families of topological vector spaces and proceeds to specialize to a number of special cases. This is either before or after he also provides a treatment of projective limits of these spaces. In each case the generality makes the context clearer and allows you to contextualize what these objects are in the grand categorical scheme of things. (...no "scheme" puns intended...)
I also would say his treatment of nuclear spaces has become outdated, even when compared to a relatively old text like the excellent survey Nuclear Locally Convex Spaces by Pietsch. At any rate, your mileage may vary, and it's far too ubiquitous a classic not to familiarize yourself with. And if you're willing to put in the time, the exercises are excellent, albeit again overly specialized and not exploring the edge cases and exceptions that I think really help flesh out one's understanding of a theory.
Displaying 1 of 1 review