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

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

is equivalent to the trivial one[1]

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

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

Examples

• Finite dimensional Banach spaces are K-spaces.
• The ${\displaystyle \ell ^{p}}$ spaces for ${\displaystyle 0<p<1}$ are K-spaces.[1]
• N. J. Kalton and N. P. Roberts proved that the Banach space ${\displaystyle \ell ^{1}}$ is not a K-space.[1]

See also

• Compactly generated space

References

1. Kalton, N. J.; Peck, N. T.; Roberts, James W. An F-space sampler. London Mathematical Society Lecture Note Series, 89. Cambridge University Press, Cambridge, 1984. xii+240 pp. ISBN 0-521-27585-7

Gelfand–Shilov space