DOI: 10.68381/jca26070 ISSN: 0944-6532
Metrizable Bounded Sets in C(X) Spaces and Distinguished C
p
(X) Spaces
Juan Carlos Ferrando, Jerzy Kąkol
Quite recently W. Ruess [Locally convex spaces not containing
\ell_{1}
ℓ
1
, Funct. Approx. Comment. Math. 50 (2014) 351–358] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's
\ell_{1}
ℓ
1
-dichotomy. Being motivated by this fact we show that for a Tychonoff space
X
X
the bounded sets of
C_{p}(X)
C
p
(
X
)
are metrizable (respectively, the bounded sets of
C_{k}(X)
C
k
(
X
)
are weakly metrizable) if and only if
X
X
is countable. If
X
X
is a
P
P
-space we show that every bounded set in
C_{p}(X)
C
p
(
X
)
is metrizable if and only if
X
X
is countable and discrete. The second part of the paper deals with distinguished
C_{p}(X)
C
p
(
X
)
spaces. Among other things we show that
C_{p}(X)
C
p
(
X
)
is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that
C_{p}(X)
C
p
(
X
)
is always distinguished whenever
X
X
is countable.