K-space_(functional_analysis) Knowpia

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 edit

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

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

References edit

^ ^a ^b ^c 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

[kalton-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

[1]

Summary

Examples edit

See also edit

References edit