In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in ...Uniform Distribution itself. so let
E
be a Banach space. then we prove:
If
F
is a bounded subset of
E
and
x
∈
co
¯
(
F
)
(= the closed convex hull of
F
), then there is a sequence (
x
n
) ⊆
F
which is Cesàro summable to
x
.
If
E
is separable,
F
⊆
E
* bounded and
f
∈
co
¯
w
∗
(
F
)
, then there is a sequence (
f
n
) ⊆
F
whose sequence of arithmetic means
f
1
+
⋯
f
N
N
,
N
≥ 1 weak*-converges to
f
.
By the aid of the Krein-Milman theorem, both (a) and (b) have interesting implications for closed, convex and bounded subsets Ω of
E
such that
Ω
=
co
¯
(
ex
Ω
)
and for weak* compact and convex subsets of
E
*. Of particular interest is the case when Ω =
B
C
(
K
)*
, where
K
is a compact metric space.
By further expanding the previous ideas and results, we are able to generalize a classical theorem of Uniform Distribution which is valid for increasing functions φ:
I
=0,1 → ℝ with φ(0) = 0 and φ(1) = 1, for functions φ of bounded variation on
I
with φ(0) = 0 and total variation
V
0
1
φ = 1.
Let
X
be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number
H
(
X
) and strict Hadwiger number
H
′
(
X
)
. More precisely, we ...define the antipodal Hadwiger number
H
α
(
X
)
as the largest cardinality of a subset
S
⊆
S
X
, such that
∀
x
≠
y
∈
S
∃
f
∈
B
X
∗
with
1
≤
f
(
x
)
-
f
(
y
)
and
f
(
y
)
≤
f
(
z
)
≤
f
(
x
)
for
z
∈
S
.
The strict antipodal Hadwiger number
H
α
′
(
X
)
is defined analogously. We prove that
H
α
′
(
X
)
=
4
for every Minkowski plane and estimate (or in some cases compute) the numbers
H
α
(
X
)
and
H
α
′
(
X
)
, where
X
=
ℓ
p
n
,
1
<
p
≤
+
∞
and
n
≥
2
. We also show that the number
H
α
′
(
X
)
grows exponentially in
dim
X
.
We show that every Banach space X has an infinite equilateral set and also that if X, then it can be renormed so as to admit an equilateral set of equal size.
Using a strengthening of the concept of Kσδ set, introduced in this paper, we study a certain subclass of the class of Kσδ Banach spaces; the so-called strongly Kσδ Banach spaces. This class of ...spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly K-analytic (SWKA) Banach spaces as the known classes of Kσδ and weakly K-analytic (WKA) Banach spaces are related.
In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in ...Uniform Distribution itself. So let \(E\) be a Banach space. Then we prove:\\ (a) If \(F\) is a bounded subset of \(E\) and \(x \in \overline{\co}(F)\) (= the closed convex hull of \(F\)), then there is a sequence \((x_n) \subseteq F\) which is Ces\`{a}ro summable to \(x\).\\ (b) If \(E\) is separable, \(F \subseteq E^*\) bounded and \(f \in \overline{\co}^{w^*}(F)\), then there is a sequence \((f_n) \subseteq F\) whose sequence of arithmetic means \(\frac{f_1+\dots+f_N}{N}\), \(N \ge 1\) weak\(^*\)-converges to \(f\). By the aid of the Krein-Milman theorem, both (a) and (b) have interesting implications for closed, convex and bounded subsets \(\Omega\) of \(E\) such that \(\Omega=\overline{\co}(\ex \Omega)\) and for weak\(^*\) compact and convex subsets of \(E^*\). Of particular interest is the case when \(\Omega=B_{C(K)^*}\), where \(K\) is a compact metric space. By further expanding the previous ideas and results, we are able to generalize a classical theorem of Uniform Distribution which is valid for increasing functions \(\varphi:I=0,1 \rightarrow \mathbb{R}\) with \(\varphi(0)=0\) and \(\varphi(1)=1\), for functions \(\varphi\) of bounded variation on \(I\) with \(\varphi(0)=0\) and total variation \(V_0^1 \varphi=1\).
Let \(X\) be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number \(H(X)\) and strict Hadwiger number \(H'(X)\). More precisely, ...we define the antipodal Hadwiger number \(H_\alpha(X)\) as the largest cardinality of a subset \(S \subseteq S_X\), such that \(\forall x \neq y \in S \,\,\, \exists f \in B_{X^*}\) with \1 \le f(x)-f(y) \,\,\, \textrm{and} \,\,\, f(y) \le f(z) \le f(x) \,\,\, \textrm{for} \,\,\, z \in S.\ The strict antipodal Hadwiger number \(H'_\alpha(X)\) is defined analogously. We prove that \(H'_\alpha(X)=4\) for every Minkowski plane and estimate (or in some cases compute) the numbers \(H_\alpha(X)\) and \(H'_\alpha(X)\), where \(X=\ell_p^n, 1 < p \le +\infty\) and \(n \ge 2\). We also show that the number \(H'_\alpha(X)\) grows exponentially in \(\dim X\).
Examples of Talagrand, Gul'ko and Corson compacta resulting from Reznichenko families of trees are presented. The
K
σ
δ
property for weakly
K
-analytic Banach spaces with an unconditional basis is ...proved.
Let \((M,d)\) be a bounded countable metric space and \(c>0\) a constant, such that \(d(x,y)+d(y,z)-d(x,z) \ge c\), for any pairwise distinct points \(x,y,z\) of \(M\). For such metric spaces we ...prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of \(\ell_\infty\).
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