For a topological space
X
X
, let
C
1
(
X
)
{C_1}(X)
denote the Banach space of all bounded functions
f
:
X
→
R
f:X \to {\mathbf {R}}
such that for every
ε
>
0
\varepsilon > 0
the set
{
x
∈
X
:
|
f
(
x
)
|
⩾
ε
}
\{ x \in X:|f(x)| \geqslant \varepsilon \}
is closed and discrete in
X
X
, endowed with the supremum norm. The main theorem is the following: Let
L
L
be a weakly countably determined subset of a Banach space; then there exist a subset
Σ
′
\Sigma ’
of the Baire space
Σ
\Sigma
, a compact space
K
K
, and a bounded linear one-to-one operator
T
:
C
(
L
)
→
C
1
(
Σ
′
×
K
)
T:C(L) \to {C_1}(\Sigma ’ \times K)
that is pointwise to pointwise continuous. In the case where
L
L
is weakly analytic,
Σ
′
\Sigma ’
can be replaced by
Σ
\Sigma
. This theorem is connected with the basic result of Amir-Lindenstrauss on WCG Banach spaces and has corresponding consequences such as: the representation of Gulko (resp. Talagrand) compact spaces as pointwise compact subsets of
C
1
(
Σ
′
×
K
)
{C_1}(\Sigma ’ \times K)
(resp.
C
1
(
Σ
×
K
)
{C_1}(\Sigma \times K)
) (a compact space
Ω
\Omega
is called Gulko or Talagrand compact if
C
(
Ω
)
C(\Omega )
is WCD or a weakly
K
K
-analytic Banach space); the characterization of WCD (resp. weakly
K
K
-analytic) Banach spaces
E
E
, using one-to-one operators from
E
∗
{E^{\ast }}
into
C
1
(
Σ
′
×
K
)
{C_1}(\Sigma ’ \times K)
(resp.
C
1
(
Σ
×
K
)
{C_1}(\Sigma \times K)
); and the existence of equivalent "good" norms on
E
E
and
E
∗
{E^{\ast }}
simultaneously.