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 ...
, an ''LF''-space, also written (''LF'')-space, is 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 ...
(TVS) ''X'' that is a locally convex
inductive limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any catego ...
of a countable inductive system
of
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
s.
This means that ''X'' is a
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of a direct system
in the category of
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
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 ...
s and each
is a Fréchet space. The name ''LF'' stands for Limit of Fréchet spaces.
If each of the bonding maps
is an embedding of TVSs then the ''LF''-space is called a strict ''LF''-space. This means that the subspace topology induced on by is identical to the original topology on .
Some authors (e.g. Schaefer) define the term "''LF''-space" to mean "strict ''LF''-space," so when reading mathematical literature, it is recommended to always check how ''LF''-space is defined.
Definition
Inductive/final/direct limit topology
Throughout, it is assumed that
*
is either the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
or some subcategory of the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of
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 ...
s (TVSs);
** If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure.
* is a non-empty
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
;
* is a family of objects in
where is a topological space for every index ;
** To avoid potential confusion, should ''not'' be called 's "initial topology" since the term "
initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...
" already has a well-known definition. The topology is called the original topology on or 's given topology.
* is a set (and if objects in
also have algebraic structures, then is automatically assumed to have has whatever algebraic structure is needed);
* is a family of maps where for each index , the map has prototype . If all objects in the category have an algebraic structure, then these maps are also assumed to be homomorphisms for that algebraic structure.
If it exists, then the
final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
on in
, also called the colimit or inductive topology in
, and denoted by or , is the
finest topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as th ...
on such that
# is an object in
, and
# for every index , the map is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
morphism in
.
In the category of topological spaces, the final topology always exists and moreover, a subset is open (resp. closed) in
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
is open (resp. closed) in for every index .
However, the final topology may
not exist in the category of
Hausdorff topological spaces due to the requirement that belong to the original category (i.e. belong to the category of Hausdorff topological spaces).
Direct systems
Suppose that is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
and that for all indices there are (continuous) morphisms in
such that if then is the identity map on and if then the following compatibility condition is satisfied:
where this means that the composition
If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set
is known as a
direct system
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from ...
in the category
that is
directed (or indexed) by . Since the indexing set is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
, the direct system is said to be directed.
The maps are called the bonding, connecting, or linking maps of the system.
If the indexing set is understood then is often omitted from the above tuple (i.e. not written); the same is true for the bonding maps if they are understood. Consequently, one often sees written " is a direct system" where "" actually represents a triple with the bonding maps and indexing set either defined elsewhere (e.g. canonical bonding maps, such as natural inclusions) or else the bonding maps are merely assumed to exist but there is no need to assign symbols to them (e.g. the bonding maps are not needed to state a theorem).
Direct limit of a direct system
For the construction of a direct limit of a general inductive system, please see the article:
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
.
Direct limits of injective systems
If each of the bonding maps
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
then the system is called injective.
If the 's have an algebraic structure, say addition for example, then for any , we pick any index such that and then define their sum using by using the addition operator of . That is,
where is the addition operator of . This sum is independent of the index that is chosen.
In the category of locally convex topological vector spaces, the topology on the direct limit of an injective directed inductive limit of locally convex spaces can be described by specifying that an
absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull ...
subset of is a neighborhood of if and only if is an absolutely convex neighborhood of in for every index .
Direct limits in Top
Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
TVSs.
In the category of topological spaces, if every bonding map is/is a
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
(resp.
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
,
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
,
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
,
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
,
quotient map
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
) then so is every .
Problem with direct limits
Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved".
For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex
nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
* Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
*Nuclear ...
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
s may to be Hausdorff (in which case the direct limit does not exist in the category of Hausdorff TVSs).
For this reason, only certain "well-behaved" direct systems are usually studied in
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. Such systems include ''LF''-spaces.
However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.
Strict inductive limit
If each of the bonding maps
is an embedding of TVSs onto proper vector subspaces and if the system is directed by
with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on .
In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .
Properties
An inductive limit in the category of locally convex TVSs of a family of
bornological (resp.
barrelled,
quasi-barrelled) spaces has this same property.
LF-spaces
Every LF-space is a
meager subset of itself.
The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete.
In particular, every LF-space is complete.
Every ''LF''-space is
barrelled and
bornological, which together with completeness implies that every LF-space is
ultrabornological.
An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.
LF space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces.
This means that ''X'' is a direct limit ...
s are
distinguished
The ruling made by the judge or panel of judges must be based on the evidence at hand and the standard binding precedents covering the subject-matter (they must be ''followed'').
Definition
In law, to distinguish a case means a court decides th ...
and their strong duals are
bornological and
barrelled (a result due to
Alexander Grothendieck).
If is the strict inductive limit of an increasing sequence of
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
then a subset of is bounded in if and only if there exists some such that is a bounded subset of .
A linear map from an LF-space into another TVS is continuous if and only if it is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
.
A linear map from an LF-space into a
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
is continuous if and only if its graph is closed in .
Every
bounded linear operator from an LF-space into another TVS is continuous.
If is an LF-space defined by a sequence
then the strong dual space
of is a Fréchet space if and only if all are
normable.
Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an
LB-space In mathematics, an ''LB''-space, also written (''LB'')-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Banach spaces.
This means that X is a direct limit of a direct sys ...
.
Examples
Space of smooth compactly supported functions
A typical example of an ''LF''-space is,
, the space of all infinitely differentiable functions on
with compact support. The ''LF''-space structure is obtained by considering a sequence of compact sets
with
and for all i,
is a subset of the interior of
. Such a sequence could be the balls of radius ''i'' centered at the origin. The space
of infinitely differentiable functions on
with compact support contained in
has a natural
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
structure and
inherits its ''LF''-space structure as described above. The ''LF''-space topology does not depend on the particular sequence of compact sets
.
With this ''LF''-space structure,
is known as the space of test functions, of fundamental importance in the
theory of distributions
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
.
Direct limit of finite-dimensional spaces
Suppose that for every positive integer , and for , consider ''X''
''m'' as a vector subspace of via the canonical embedding defined by .
Denote the resulting LF-space by . Since any TVS topology on makes continuous the inclusions of the ''X''
''m'''s into , the latter space has the maximum among all TVS topologies on an
-vector space with countable
Hamel dimension
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
. It is a LC topology, associated with the family of all seminorms on . Also, the TVS inductive limit topology of coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces in the category TOP and in the category TVS coincide. The continuous dual space
of is equal to the
algebraic dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of , that is the space of all real valued sequences
and the weak topology on
is equal to the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
* the final topology on the disjoint union
* the to ...
on
(i.e.
).. In fact, it is the unique LC topology on
whose topological dual space is X.
Furthermore, the canonical map of into the continuous dual space of
is surjective.
See also
*
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
*
Direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
*
Final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
*
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that
# Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex.
...
*
LB-space In mathematics, an ''LB''-space, also written (''LB'')-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Banach spaces.
This means that X is a direct limit of a direct sys ...
Citations
Bibliography
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{{TopologicalVectorSpaces
Topological vector spaces