  This paper is intended to provide foundations to the theory of Witt-type
topological group and ring functors defined on a category of topological
algebras, and, in presence of Banach norms, to show how to topologically deal
with them. It is logically the first of a series of papers in preparation on
the use of Barsotti-Witt constructions to obtain Scholze's tilting equivalence
uniformly with respect to the perfectoid field K of characteristic 0 lifting a
particular perfectoid field F of characteristic p>0.
  The paper is basically self-contained and may have an independent interest
especially for specialists of topological algebra and non-archimedean
functional analysis: this accounts for its independent submission. We indicate
a new viewpoint in the theory of non-archimedean Banach algebras, based on a
higher-dimensional generalization of the notion of gauge-seminorm as explained
in P. Schneider "Non-archimedean Functional Analysis" Springer 2002
