K-space (functional analysis)

In mathematics, more specifically in functional analysis, a K-space is an F-space ${\displaystyle V}$such that every extension of F-spaces (or twisted sum) of the form $${\displaystyle 0\rightarrow \mathbb {R} \rightarrow X\rightarrow V\rightarrow 0.\,\!}$$is equivalent to the trivial one $${\displaystyle 0\rightarrow \mathbb {R} \rightarrow \mathbb {R} \times V\rightarrow V\rightarrow 0.\,\!}$$where ${\displaystyle \mathbb {R} }$is the real line.

The ${\displaystyle \ell ^{p}}$spaces for ${\displaystyle 0<p<1}$are K-spaces, as are all finite dimensional Banach spaces.

N. J. Kalton and N. P. Roberts proved that the Banach space ${\displaystyle \ell ^{1}}$is not a K-space.

• Compactly generated space – Property of topological spaces
• Gelfand–Shilov space