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.
