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

$0\rightarrow \mathbb {R} \rightarrow X\rightarrow V\rightarrow 0.\,\!$

is equivalent to the trivial one

$0\rightarrow \mathbb {R} \rightarrow \mathbb {R} \times V\rightarrow V\rightarrow 0.\,\!$

where $\mathbb {R}$ is the real line.

Examples

• Finite dimensional Banach spaces are K-spaces.
• The $\ell ^{p}$ spaces for $0<p<1$ are K-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

References

Gelfand–Shilov space