In mathematics, a p-adic distribution is an analogue of ordinary
distributions (i.e. generalized functions) that takes values in a ring of
''p''-adic numbers.
Definition
If ''X'' is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, a distribution on ''X'' with values in an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''G'' is a finitely additive function from the
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
open subsets of ''X'' to ''G''. Equivalently, if we define the space of
test functions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
to be the locally constant and compactly supported integer-valued functions, then a distribution is an additive map from test functions to ''G''. This is formally similar to the usual definition of distributions, which are continuous linear maps from a space of test functions on a manifold to the real numbers.
''p''-adic measures
A ''p''-adic measure is a special case of a ''p''-adic distribution, analogous to a measure on a measurable space. A ''p''-adic distribution taking values in a normed space is called a ''p''-adic measure if the values on compact open subsets are bounded.
References
*
*
*
*{{Citation , last1=Washington , first1=Lawrence C. , title=Cyclotomic fields , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , edition=2nd , isbn=978-0-387-94762-4 , year=1997
Number theory
p-adic numbers