In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Kolmogorov's normability criterion is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
that provides a
necessary and sufficient condition for a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
to be ; that is, for the existence of a
norm on the space that generates the given
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
The normability criterion can be seen as a result in same vein as the
Nagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite ( ...
and
Bing metrization theorem
In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Formal statement
The theorem states that a topological space X is metrizable if and only if it is regular and T0 and h ...
, which gives a necessary and sufficient condition for 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 poin ...
to be
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
. The result was proved by the Russian mathematician
Andrey Nikolayevich Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
in 1934.
[ (See Section 8.1.3)]
Statement of the theorem
Because translation (that is, vector addition) by a constant preserves the convexity, boundedness, and
openness of sets, the words "of the origin" can be replaced with "of some point" or even with "of every point".
Definitions
It may be helpful to first recall the following terms:
* A (TVS) is a vector space
equipped with a topology
such that the vector space operations of scalar multiplication and vector addition are continuous.
* A topological vector space
is called if there is a
norm on
such that the open balls of the norm
generate the given topology
(Note well that a given normable topological vector space might admit multiple such norms.)
* 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 poin ...
is called a if, for every two distinct points
there is an open
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of
that does not contain
In a topological vector space, this is equivalent to requiring that, for every
there is an open neighbourhood of the origin not containing
Note that being T
1 is weaker than being a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
, in which every two distinct points
admit open neighbourhoods
of
and
of
with
; since normed and normable spaces are always Hausdorff, it is a "surprise" that the theorem only requires T
1.
* A subset
of a vector space
is a if, for any two points
the line segment joining them lies wholly within
that is, for all
* A subset
of a topological vector space
is a if, for every open neighbourhood
of the origin, there exists a scalar
so that
(One can think of
as being "small" and
as being "big enough" to inflate
to cover
)
See also
*
*
*
References
{{Topological vector spaces
Theorems in functional analysis