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 $${\displaystyle 0\to \mathbb{ℝ}\to X\to V\to 0.}$$ is equivalent to the trivial one^[1] $${\displaystyle 0\to \mathbb{ℝ}\to \mathbb{ℝ}\times V\to V\to 0.}$$ where ${\displaystyle \mathbb{ℝ}}$ is the real line.

The ${\displaystyle {\ell }^p}$ spaces for ${\displaystyle 0<p<1}$ are K-spaces,^[1] 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.^[1]

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

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