View source for K-space (functional analysis)

In [[Mathematics|mathematics]], more specifically in [[Functional analysis|functional analysis]], a '''K-space''' is an [[F-space]] <math>V</math> such that every extension of F-spaces (or twisted sum) of the form
<math display=block>0 \rightarrow \R \rightarrow X \rightarrow V \rightarrow 0. \,\!</math>
is equivalent to the trivial one<ref name="kalton">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}}</ref>
<math display=block>0\rightarrow \R \rightarrow \R \times V \rightarrow V \rightarrow 0. \,\!</math>
where <math>\R</math> is the [[Real line|real line]].

==Examples==

The [[Lp space|<math>\ell^p</math> spaces]] for <math>0< p < 1</math> are K-spaces,<ref name="kalton"/> as are all finite dimensional [[Banach space]]s.

N. J. Kalton and N. P. Roberts proved that the Banach space <math>\ell^1</math> is not a K-space.<ref name="kalton"/>

==See also==

* {{annotated link|Compactly generated space}}
* {{annotated link|Gelfand–Shilov space}}

==References==

{{reflist|group=note}}
{{reflist}}

{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:F-spaces]]
[[Category:Topological vector spaces]]

{{Sourceattribution|K-space (functional analysis)}}