K-space (functional analysis)

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

Examples

The [math]\displaystyle{ \ell^p }[/math] spaces for [math]\displaystyle{ 0\lt p \lt 1 }[/math] are K-spaces, as are all finite dimensional Banach spaces.

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

See also

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