DF-space

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, DF-spaces, also written (DF)-spaces are <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> having a property that is shared by locally convex <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable topological vector spaces</a>. They play a considerable part in the theory of topological tensor products.
DF-spaces were first defined by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> and studied in detail by him in (Grothendieck 1954). 
Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> locally convex space and 
 
 
 
 
 V
 
 1
 
 
 ,
 
 V
 
 2
 
 
 ,
 …
 
 
 {\displaystyle V_{1},V_{2},\ldots }
 
 is a sequence of convex 0-neighborhoods in 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 such that 
 
 
 
 V
 :=
 
 ∩
 
 i
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V:=\cap _{i}V_{i}}
 
 absorbs every strongly bounded set, then 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a 0-neighborhood in 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 (where 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 is the continuous dual space of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 endowed with the strong dual topology).

DF-space open-in-new

DF-space