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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.

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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).

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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.

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