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^[1]

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

References

1 2 3 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

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[kalton-1] 1 2 3 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