Bibliography for topologies defined by a family of seminorms 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.
 A: 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). 
A: 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


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


II Duality, Spaces of Distributions


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


III Tensor Products. Kernels


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*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. 
