Let
U
U
be an open subset of a locally convex space
E
E
, and let
H
c
(
U
,
F
)
{H_c}(U,F)
denote the vector space of holomorphic functions into a locally convex space
F
F
, endowed with continuous convergence. It is shown that if
F
F
is a semi-Montel space, then the bounded subsets of
H
c
(
U
,
F
)
{H_c}(U,F)
are relatively compact. Further it is shown that
E
E
is a Schwartz space iff the continuous convergence structure on the algebra
H
(
U
)
H(U)
of scalar-valued holomorphic functions on
U
U
, coincides with local uniform convergence. Using this, an example of a nuclear Fréchet space
E
E
is given, such that the locally convex topology associated with continuous convergence on
H
(
E
)
H(E)
is strictly finer than the compact open topology. Thus, the behavior of the space
H
c
(
E
)
{H_c}(E)
differs in this respect from that of its subspace
L
c
E
{L_c}E
of linear forms and that of its superspace
C
c
(
E
)
{C_c}(E)
of continuous functions.