In mathematics, an ** LF-space**, also written

- Definition
- Inductive/final/direct limit topology
- Direct systems
- Direct limit of a direct system
- Strict inductive limit
- Properties
- LF-spaces
- Examples
- Space of smooth compactly supported functions
- Direct limit of finite-dimensional spaces
- See also
- Citations
- Bibliography

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

Throughout, it is assumed that

- is either the category of topological spaces or some subcategory of the category of topological vector spaces (TVSs);
- If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure.

- I is a non-empty directed set;
*X*_{•}= (*X*_{i})_{i ∈ I}is a family of objects in where (*X*_{i}, τ_{Xi}) is a topological space for every index i;- To avoid potential confusion, τ
_{Xi}shouldbe called**not***X*_{i}'s "initial topology" since the term "initial topology" already has a well-known definition. The topology τ_{Xi}is called the**original**topology on*X*_{i}or*X*_{i}'s**given topology**.

- To avoid potential confusion, τ
- X is a set (and if objects in also have algebraic structures, then X is automatically assumed to have has whatever algebraic structure is needed);
*f*_{•}= (*f*_{i})_{i ∈ I}is a family of maps where for each index i, the map has prototype*f*_{i}: (*X*_{i}, τ_{Xi}) →*X*. 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 on X in **, also called the **colimit** or **inductive topology** in , and denoted by τ_{f•} or τ_{f}, is the finest topology on X such that

- (
*X*, τ_{f}) is an object in , and - for every index i, the map
*f*_{i}: (*X*_{i}, τ_{Xi}) → (*X*, τ_{f}) is a continuous morphism in .

In the category of topological spaces, the final topology always exists and moreover, a subset *U* ⊆ *X* is open (resp. closed) in (*X*, τ_{f}) if and only if *f* _{i}^{- 1} (*U*) is open (resp. closed) in (*X*_{i}, τ_{Xi}) for every index i.

However, the final topology may *not* exist in the category of Hausdorff topological spaces due to the requirement that (*X*, τ_{Xf}) belong to the original category (i.e. belong to the category of Hausdorff topological spaces).^{ [3] }

Suppose that (*I*, ≤) is a directed set and that for all indices *i* ≤ *j* there are (continuous) morphisms in

such that if *i* = *j* then *f* _{i}^{j} is the identity map on *X*_{i} and if *i* ≤ *j* ≤ *k* 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 the category that is **directed** (or **indexed**) by *I*. Since the indexing set I is a directed set, the direct system is said to be **directed**.^{ [4] } The maps *f* _{i}^{j} are called the **bonding**, **connecting**, or **linking****maps** of the system.

If the indexing set I is understood then I 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 "*X*_{•} is a direct system" where "*X*_{•}" 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).

For the construction of a direct limit of a general inductive system, please see the article: direct limit.

**Direct limits of injective systems**

If each of the bonding maps is injective then the system is called **injective**.^{ [4] }

In^{j}

_{i} : *X*_{i} → *X*_{j}

(i.e. defined by *x* ↦ *x*) so that that the subspace topology on *X*_{i} induced by *X*_{j} is weaker (i.e. coarser) than the original (i.e. given) topology on *X*_{i}.

In this case, also take

If the *X*_{i}'s have an algebraic structure, say addition for example, then for any *x*, *y* ∈ *X*, we pick any index *i* such that *x*, *y* ∈ *X*_{i} and then define their sum using by using the addition operator of *X*_{i}. That is,

where +_{i} is the addition operator of *X*_{i}. This sum is independent of the index i that is chosen.

In the category of locally convex topological vector spaces, the topology on the direct limit X of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset U of X is a neighborhood of 0 if and only if *U* ∩ *X*_{i} is an absolutely convex neighborhood of 0 in *X*_{i} for every index i.^{ [4] }

**Direct limits in Top**

Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and locally convex TVSs. In the category of topological spaces, if every bonding map *f* _{i}^{j} is/is a injective (resp. surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every *f*_{i} : *X*_{i} → *X*.^{ [3] }

Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved".^{ [4] } For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex nuclear Fréchet spaces may **fail** 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. Such systems include *LF*-spaces.^{ [4] } However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.^{ [4] }

If each of the bonding maps is an embedding of TVSs onto proper vector subspaces and if the system is directed by ℕ 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 *X*_{i} is a vector subspace of *X*_{i+1} and that the subspace topology induced on *X*_{i} by *X*_{i+1} is identical to the original topology on *X*_{i}.^{ [1] }

In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces X can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if *U* ∩ *X*_{n} is an absolutely convex neighborhood of 0 in *X*_{n} for every n.

An inductive limit in the category of locally convex TVSs of a family of bornological (resp. barrelled, quasi-barrelled) spaces has this same property.^{ [5] }

Every LF-space is a meager subset of itself.^{ [6] } 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.^{ [7] } 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.^{ [8] } LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).

If X is the strict inductive limit of an increasing sequence of Fréchet space *X*_{n} then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of *X*_{n}.^{ [7] }

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.^{ [9] } A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in *X*×*Y*.^{ [10] } Every bounded linear operator from an LF-space into another TVS is continuous.^{ [11] }

If X is an LF-space defined by a sequence then the strong dual space of X is a Fréchet space if and only if all *X*_{i} are normable.^{ [12] } Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.

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 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.

Suppose that for every positive integer n, *X*_{n} := ℝ^{n} and for *m* < *n*, consider *X*_{m} as a vector subspace of *X*_{n} via the canonical embedding *X*_{m} → *X*_{n} defined by *x* := (*x*_{1}, ..., *x*_{m}) ↦ (*x*_{1}, ..., *x*_{m}, 0, ..., 0). Denote the resulting LF-space by X. The continuous dual space of X is equal to the algebraic dual space of X and the weak topology on is equal to the strong topology on (i.e. ).^{ [13] } Furthermore, the canonical map of X into the continuous dual space of is surjective.^{ [13] }

- 1 2 3 Schaefer & Wolff 1999, pp. 55-61.
- ↑ Helgason, Sigurdur (2000).
*Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions*(Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5. - 1 2 Dugundji 1966, pp. 420-435.
- 1 2 3 4 5 6 Bierstedt 1988, pp. 41-56.
- ↑ Grothendieck 1973, pp. 130-142.
- ↑ Narici & Beckenstein 2011, p. 435.
- 1 2 Schaefer & Wolff 1999, pp. 59-61.
- ↑ Narici & Beckenstein 2011, p. 436.
- ↑ Trèves 2006, p. 141.
- ↑ Trèves 2006, p. 173.
- ↑ Trèves 2006, p. 142.
- ↑ Trèves 2006, p. 201.
- 1 2 Schaefer & Wolff 1999, p. 201.

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In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

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. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

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 vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector space (TVS) 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 that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In functional analysis and related areas of mathematics, a **Montel space**, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In general topology and related areas of mathematics, the **final topology** on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

In mathematics, a **nuclear space** is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, particularly in functional analysis, a **webbed space** is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a *web* that satisfies certain properties. Webs were first investigated by de Wilde.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map continuous is called the **projective topology** or the **π-topology**. When is endowed with this topology then it is denoted by and called the **projective tensor product** of and

In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

In mathematics, an ** LB-space**, also written

In functional analysis, a topological vector space (TVS) is called **ultrabornological** if every bounded linear operator from into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck.

In the mathematical discipline of functional analysis, **differentiable vector-valued functions from Euclidean space** are differentiable TVS-valued functions whose domains are subset of finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.

In functional analysis, a **topological homomorphism** or simply **homomorphism** is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

**F. Riesz's theorem** is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

In mathematics, particularly in functional analysis and topology, the **closed graph theorem** is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In mathematical analysis, the **spaces of test functions and distributions** are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the *canonical LF-topoogy*, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called * the space of distributions on * and is denoted by where the "" subscript indicates that the continuous dual space of denote by is endowed with the strong dual topology.

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