Math 2.0

Functional analysis Definition

LF-space

Builds on Fréchet spaceLocally convex spaceConvex setBalanced setNeighbourhood

Definition

Let 𝕂\mathbb{K} be ℝ\R or ℂ\C, and let EE be a vector space over 𝕂\mathbb{K}. Suppose E=⋃n≥1EnE = \bigcup_{n \ge 1} E_n, where E1⊆E2⊆⋯E_1 \subseteq E_2 \subseteq \cdots are linear subspaces. Suppose each EnE_n carries a topology making it a Fréchet space, and each inclusion En→En+1E_n \to E_{n+1} is continuous. The inductive limit topology on EE is the locally convex topology defined as follows. A set is a neighbourhood of 00 if it contains a convex balanced set UU with the following property. For every nn, the set U∩EnU \cap E_n is a neighbourhood of 00 in EnE_n. A locally convex space is an LF-space if it is Hausdorff and arises in this way.

Remarks

The inductive limit topology is the finest locally convex topology on EE for which all inclusions En→EE_n \to E are continuous. Many authors require the limit to be strict: each EnE_n carries the topology induced by En+1E_{n+1}. Consider a strict limit with each EnE_n closed in En+1E_{n+1}. Then EE is Hausdorff and complete, and it induces the given topology on each EnE_n. A set is bounded in EE if and only if it lies in some EnE_n and is bounded there. Example: let Ω⊆ℝd\Omega \subseteq \R^d be open, and let K1⊆K2⊆⋯K_1 \subseteq K_2 \subseteq \cdots be compact sets with union Ω\Omega, each inside the interior of the next. Let 𝒟Kn\mathcal{D}_{K_n} be the Fréchet space of smooth functions supported in KnK_n. Then Cc∞(Ω)=⋃n𝒟KnC_c^\infty(\Omega) = \bigcup_n \mathcal{D}_{K_n} is a strict LF-space. Non-example: the space c00c_{00} of finitely supported sequences with the sup norm is not an LF-space, since every LF-space is barrelled and this space is not.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Balanced set, concept, no evidence levelBalanced setconceptConvex set, concept, no evidence levelConvex setconceptFréchet space, concept, no evidence levelFréchet spaceconceptLocally convex space, concept, no evidence levelLocally convex spaceconceptNeighbourhood, concept, no evidence levelNeighbourhoodconceptLF-space, definition, no evidence levelLF-spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Balanced set concept
  2. Convex set concept
  3. Fréchet space concept
  4. Locally convex space concept
  5. Neighbourhood concept
  6. LF-space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/LF-space
About this page
TagHJ2QSEMHMC
URIm2://HJ2QSEMHMC
Kinddefinition
WritingW0 stub
MSC46A13
DefinesLF-space
UsesFréchet space, Locally convex space, Convex set, Balanced set, Neighbourhood
Changed2 November 2026

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