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research
Uncountable sets of unit vectors that are separated by more than 1
Authors
Tomasz Kania
Tomasz Kochanek
Publication date
1 January 2016
Publisher
'Institute of Mathematics, Polish Academy of Sciences'
Doi
Cite
View
on
arXiv
Abstract
Let
X
X
X
be a Banach space. We study the circumstances under which there exists an uncountable set
A
⊂
X
\mathcal A\subset X
A
⊂
X
of unit vectors such that
∥
x
−
y
∥
>
1
\|x-y\|>1
∥
x
−
y
∥
>
1
for distinct
x
,
y
∈
A
x,y\in \mathcal A
x
,
y
∈
A
. We prove that such a set exists if
X
X
X
is quasi-reflexive and non-separable; if
X
X
X
is additionally super-reflexive then one can have
∥
x
−
y
∥
⩾
1
+
ε
\|x-y\|\geqslant 1+\varepsilon
∥
x
−
y
∥
⩾
1
+
ε
for some
ε
>
0
\varepsilon>0
ε
>
0
that depends only on
X
X
X
. If
K
K
K
is a non-metrisable compact, Hausdorff space, then the unit sphere of
X
=
C
(
K
)
X=C(K)
X
=
C
(
K
)
also contains such a subset; if moreover
K
K
K
is perfectly normal, then one can find such a set with cardinality equal to the density of
X
X
X
; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat
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Last time updated on 21/03/2020