Bibliography for topologies defined by a family of seminorms - MathOverflow

Bibliography for topologies defined by a family of seminorms

2009-12-22T12:25:37Z

Hello

I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.

Thank you very much.

Answer by Xabier Domínguez for Bibliography for topologies defined by a family of seminorms

2009-12-22T18:16:59Z

I don't know much about distributions but if you're entering this area with such a motivation, maybe you could use Horvath's book "Topological vector spaces and distributions". The first part is a fine introduction to locally convex space theory in itself, and the presentation of this (rather standard) material should be convenient for anybody interested on the final chapter -- distributions.

By the way, a linear topology on a vector space which is defined using a family of seminorms is in general a locally convex topology, not necessarily metrizable (Fréchet spaces are metrizable and complete locally convex spaces).

Answer by Ulrich Pennig for Bibliography for topologies defined by a family of seminorms

2010-03-12T10:56:25Z

I just had a look at

Topological Vector Spaces, Distributions and Kernels

by Francois Treves. It is divided into three parts:

I Topological Vector Spaces. Spaces of Funtions

covering: basic material about locally convex spaces and Frechet spaces (with a lot of examples)

II Duality, Spaces of Distributions

topologies on Duals, transposes of linear maps, convolution, barreled spaces

III Tensor Products. Kernels

injective and projective tensor products and their relation to bilinear forms, nuclear spaces, nuclear mappings, Schwartz kernel theorem and applications

From the first sight, this looks like a good place to start if you are already familiar with functional analysis on Banach and Hilbert spaces.