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I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points):

$\bullet$ Seminorms

$\bullet$ Locally convex topological vector spaces

$\bullet$ weak-$^{*}$ topology

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    $\begingroup$ You probably want to look for a book on topological vector spaces, not general topology. $\endgroup$ – JHance Jul 11 '14 at 14:32
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Check out http://www.math.uwaterloo.ca/~lwmarcou/Preprints/LinearAnalysis.pdf

This was the book used for my functional analysis course. It is a functional text more than a topology text book, but has chapters on those 3 bullets you listed and other related topics. I liked it very much.

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