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

Examples

The ℓ p {\displaystyle \ell ^{p}} spaces for 0 < p < 1 {\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 ℓ 1 {\displaystyle \ell ^{1}} is not a K-space.

See also