K-space (functional analysis) explained

V such that every extension of F-spaces (or twisted sum) of the form$$0\; \backslash rightarrow\; \backslash R\; \backslash rightarrow\; X\; \backslash rightarrow\; V\; \backslash rightarrow\; 0.\; \backslash ,\backslash !$$is equivalent to the trivial one^[1] $$0\backslash rightarrow\; \backslash R\; \backslash rightarrow\; \backslash R\; \backslash times\; V\; \backslash rightarrow\; V\; \backslash rightarrow\; 0.\; \backslash ,\backslash !$$where \R is the real line.

Examples

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]

Notes and 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.