LF-space

If each of the bonding maps ${\displaystyle i_{nm}}$ is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on X_n by X_n+1 is identical to the original topology on X_n. 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

• ${\displaystyle \mathcal{C}}$ 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 ${\displaystyle \mathcal{C}}$ where (X_i, τ_Xi) is a topological space for every index i;
  • To avoid potential confusion, τ_Xi should not be called 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.
• X is a set (and if objects in ${\displaystyle \mathcal{C}}$ also have algebraic structures, then X is automatically assumed to have 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 ${\displaystyle \mathcal{C}}$, also called the colimit or inductive topology in ${\displaystyle \mathcal{C}}$, and denoted by τ_f• or τ_f, is the finest topology on X such that

1. (X, τ_f) is an object in ${\displaystyle \mathcal{C}}$, and
2. for every index i, the map f_i : (X_i, τ_Xi) → (X, τ_f) is a continuous morphism in ${\displaystyle \mathcal{C}}$.

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).

Direct systems

Suppose that (I, ≤) is a directed set and that for all indices i ≤ j there are (continuous) morphisms in ${\displaystyle \mathcal{C}}$

f_i^j : X_i → X_j

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:

f_i^k = f_j^k ∘ f_i^j,

where this means that the composition

${\displaystyle X_i\stackrel{f_i^j}{\to }{X}_j\stackrel{f_j^k}{\to }{X}_k\text{ is equal to }{X}_i\stackrel{f_i^k}{\to }{X}_k.}$

If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set

${\displaystyle \left(X_{\bullet },\left\{f_i^j:i,j\in I\text{ and }i\le j\right\},I\right)}$

is known as a direct system in the category ${\displaystyle \mathcal{C}}$ that is directed (or indexed) by I. Since the indexing set I is a directed set, the direct system is said to be directed. 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).

Direct limit of a direct system

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 ${\displaystyle f_i^j}$ is injective then the system is called injective.

Assumptions: In the case where the direct system is injective, it is often assumed without loss of generality that for all indices i ≤ j, each X_i is a vector subspace of X_j (in particular, X_i is identified with the range of ${\displaystyle f_i^j}$) and that the bonding map ${\displaystyle f_i^j}$ is the natural inclusion In^j
_i : X_i → X_j

(i.e. defined by x ↦ x) so 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

X := X_i. The limit maps are then the natural inclusions In_i : X_i → X. The direct limit topology on X is the final topology induced by these inclusion maps.

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,

x + y := x +_iy,

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.

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.

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 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. However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.

Strict inductive limit

If each of the bonding maps ${\displaystyle f_i^j}$ is an embedding of TVSs onto proper vector subspaces and if the system is directed by ${\displaystyle \mathbb{N}}$ 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.

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.