LF-space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an LF-space, also written (LF)-space, is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) X that is a locally convex <a href="/facts/Inductive_limit/V3JuHnYK">inductive limit</a> of a countable inductive system 
 
 
 
 (
 
 X
 
 n
 
 
 ,
 
 i
 
 n
 m
 
 
 )
 
 
 {\displaystyle (X_{n},i_{nm})}
 
 of <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>. 
This means that X is a <a href="/facts/Direct_limit/V3JuHnYK">direct limit</a> of a direct system 
 
 
 
 (
 
 X
 
 n
 
 
 ,
 
 i
 
 n
 m
 
 
 )
 
 
 {\displaystyle (X_{n},i_{nm})}
 
 in the category of <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> and each 
 
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle X_{n}}
 
 is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
If each of the bonding maps 
 
 
 
 
 i
 
 n
 m
 
 
 
 
 {\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 Xn by Xn+1 is identical to the original topology on Xn.
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.

LF-space open-in-new

LF-space