K-space (functional analysis)

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

Examples

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

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

See also

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