For any compact Hausdorff spaces
X
,
Y
X,Y
with
φ
:
X
→
Y
\varphi :\;X \to Y
a continuous onto mapping,
E
,
F
E,F
, Hausdorff locally convex spaces with
F
F
complete,
C
(
X
,
E
)
(
C
(
Y
,
E
)
)
C(X,E)\;(C(Y,E))
all
E
E
-valued continuous functions on
X
(
Y
)
X(Y)
, and
L
:
C
(
Y
,
E
)
→
F
L:C(Y,E) \to F
a
T
\mathcal {T}
-compact continuous operator
(
σ
(
F
,
F
′
)
≤
T
≤
τ
(
F
,
F
′
)
)
(\sigma (F,F’) \leq \mathcal {T} \leq \tau (F,F’))
, it is proved there exists a
T
\mathcal {T}
-compact continuous operator
L
0
:
C
(
X
,
E
)
→
F
{L_0}:C(X,E) \to F
such that
L
0
(
f
∘
φ
)
=
L
(
f
)
{L_0}(f \circ \varphi ) = L(f)
for every
f
∈
C
(
Y
,
E
)
f \in C(Y,E)
.