Montel space
Encyclopedia
In functional analysis
and related areas of mathematics
a Montel space, named after Paul Montel, is any topological vector space
in which an analog of Montel's theorem
holds. Specifically, a Montel space is a barrelled
topological vector space
where every closed
and bounded set
is compact
. That is, it satisfies the Heine-Borel property.
In classical complex analysis
, Montel's theorem asserts that the space of holomorphic function
s on an open
connected
subset of the complex number
s has this property.
Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions
. The space C∞(Ω) of smooth function
s on an open set Ω in Rn is a Montel space equipped with the topology induced by the family of seminorms
for n = 1,2,… and K ranges over compact subsets of Ω, and α is a multi-index. Similarly, is the space of compactly supported functions in an open set with the final topology
of the family of inclusions
as K ranges over all compact subsets of Ω. The Schwartz space is also a Montel space.
No infinite-dimensional Banach space
is a Montel space, since these cannot satisfy the Heine-Borel property: the closed unit ball is closed and bounded, but not compact.
Strong duals of Montel spaces are Montel Spaces.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a Montel space, named after Paul Montel, is any topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
in which an analog of Montel's theorem
Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal....
holds. Specifically, a Montel space is a barrelled
Barrelled space
In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
where every closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and bounded set
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. That is, it satisfies the Heine-Borel property.
In classical complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, Montel's theorem asserts that the space of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on an open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
subset of the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s has this property.
Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
. The space C∞(Ω) of smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s on an open set Ω in Rn is a Montel space equipped with the topology induced by the family of seminorms
for n = 1,2,… and K ranges over compact subsets of Ω, and α is a multi-index. Similarly, is the space of compactly supported functions in an open set with the final topology
Final topology
In general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...
of the family of inclusions
as K ranges over all compact subsets of Ω. The Schwartz space is also a Montel space.No infinite-dimensional Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
is a Montel space, since these cannot satisfy the Heine-Borel property: the closed unit ball is closed and bounded, but not compact.
Strong duals of Montel spaces are Montel Spaces.