  Roughly spoken, Functionalanalysis means the study of the category of
infinite-dimensional vectorspaces over the field of real or complex numbers,
together with their linear maps. In most cases, one further needs a topological
structure on such a vectorspace, because then, you can consider the continuous
linear maps between such spaces. The name Functionalanalysis is due to the
fact, that in the beginning of the theory, the authors wanted to expand
Calculus onto functionals of spaces of functions. Functionalanalytical results
give the possibility to solve problems in the Theory of (Partial) Differential
Equations, in Complex Analysis or in Quantum Mechanics. But the aim of this
lines is not to explain the applications. We will discuss the mathematical
theory of almost metric spaces, normed vector spaces and algebras, spaces of
continuous resp. $p$-integrable functions as well as reflexive and uniformly
convex spaces.
  ad v2: We added that, in the case $p \in {]}0,1{[}$, $L^p$ is the completion
of the compactly supported continuous functions (with the obvious metric), too.
Actually, the proof is the same as in the case $p \in {[}1, \infty{[}$.
