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In
mathematics , more specifically in
functional analysis , a K-space is an
F-space
V
{\displaystyle V}
such that every extension of F-spaces (or twisted sum) of the form
0
→
R
→
X
→
V
→
0.
{\displaystyle 0\rightarrow \mathbb {R} \rightarrow X\rightarrow V\rightarrow 0.\,\!}
is equivalent to the trivial one
[1]
0
→
R
→
R
×
V
→
V
→
0.
{\displaystyle 0\rightarrow \mathbb {R} \rightarrow \mathbb {R} \times V\rightarrow V\rightarrow 0.\,\!}
where
R
{\displaystyle \mathbb {R} }
is the
real line .
Examples
The
ℓ
p
{\displaystyle \ell ^{p}}
spaces for
0
<
p
<
1
{\displaystyle 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
ℓ
1
{\displaystyle \ell ^{1}}
is not a K-space.
[1]
See also
References
^
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
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