This book grew out of a course which I gave during the winter term
1997/98 at the Universitat Munster. The course covered the material
which here is presented in the first three chapters. The fourth
more advanced chapter was added to give the reader a rather
complete tour through all the important aspects of the theory of
locally convex vector spaces over nonarchimedean fields. There is
one serious restriction, though, which seemed inevitable to me in
the interest of a clear presentation. In its deeper aspects the
theory depends very much on the field being spherically complete or
not. To give a drastic example, if the field is not spherically
complete then there exist nonzero locally convex vector spaces
which do not have a single nonzero continuous linear form. Although
much progress has been made to overcome this problem a really nice
and complete theory which to a large extent is analogous to
classical functional analysis can only exist over spherically
complete field8. I therefore allowed myself to restrict to this
case whenever a conceptual clarity resulted. Although I hope that
thi8 text will also be useful to the experts as a reference my own
motivation for giving that course and writing this book was
different. I had the reader in mind who wants to use locally convex
vector spaces in the applications and needs a text to quickly gra8p
this theory.
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