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 $(X_{n},i_{nm})$ of Fréchet spaces.^[1] This means that X is a direct limit of a direct system $(X_{n},i_{nm})$ in the category of locally convex topological vector spaces and each $X_{n}$ is a Fréchet space. The name LF stands for Limit of Fréchet spaces.

If each of the bonding maps $i_{nm}$ 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 $X n$ .^[1]^[2] 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.

^ ^a ^b 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.

[FOOTNOTESchaeferWolff199955–61-1] Schaefer & Wolff 1999, pp. 55–61.

[2] 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.

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