In mathematics, more specifically in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, a K-space is an

F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...

V

such that every extension of F-spaces (or twisted sum) of the form

0 \rightarrow \R \rightarrow X \rightarrow V \rightarrow 0. \,\!

is equivalent to the trivial oneKalton, 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.

0\rightarrow \R \rightarrow \R \times V \rightarrow V \rightarrow 0. \,\!

where

\R

is the real line.

Examples

The

\ell^p

spaces for

0< p < 1

are K-spaces, 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.

References

{{Topological vector spaces Functional analysis F-spaces Topological vector spaces

Examples

See also

References