Abstract
AbstractThe linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space $$\mathbb {R}^X$$
R
X
) is denoted by $$C_p(X).$$
C
p
(
X
)
.
In this paper we continue the systematic study of sequences spaces $$c_{0}$$
c
0
and $$\ell _{q}$$
ℓ
q
(for $$0<q\le \infty $$
0
<
q
≤
∞
) with the topology induced from $$\mathbb {R}^{\mathbb {N}}$$
R
N
(denoted by $$(c_{0})_p$$
(
c
0
)
p
and $$(\ell _{q})_{p}$$
(
ℓ
q
)
p
, respectively) and their role in the theory of $$C_p(X)$$
C
p
(
X
)
spaces. For every infinite Tychonoff space X we construct a subspace F of $$C_p(X)$$
C
p
(
X
)
that is isomorphic to $$(c_{0})_p$$
(
c
0
)
p
; if X contains an infinite compact subset, then the copy F of $$(c_{0})_p$$
(
c
0
)
p
is closed in $$C_p(X)$$
C
p
(
X
)
. It follows that $$C_p(X)$$
C
p
(
X
)
contains a copy of $$(\ell _{q})_{p}$$
(
ℓ
q
)
p
for every $$0<q\le \infty $$
0
<
q
≤
∞
. We prove that for any infinite compact space X the space $$C_p(X)$$
C
p
(
X
)
contains no closed copy of $$(\ell _{q})_{p}$$
(
ℓ
q
)
p
for $$q\in (0, 1]\cup \{\infty \}$$
q
∈
(
0
,
1
]
∪
{
∞
}
and no complemented copy for $$0<q\le \infty $$
0
<
q
≤
∞
. Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference42 articles.
1. Arkhangel’ski, A.V.: Topological Function Spaces. Kluwer, Dordrecht (1992)
2. Arkhangel’ski, A. V.: $$C_p$$-theory, in: Recent Progress in General Topology, North-Holland, 1-56 (1992)
3. Banakh, T., Ka̧kol, J.: Metrizable quotients of $$C_p$$-spaces. Topol. Appl. 249, 95–102 (2018)
4. Banakh, T., Ka̧kol, J., Śliwa, W.: Josefson-Nissenzweig property for $$C_{p}$$-spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 3015–3030 (2019)
5. Cembranos, P.: $$C(K, E)$$ contains a complemented copy of $$c_{0}$$. Proc. Am. Math. Soc. 91, 556–558 (1984)