Montel's Theorem
In
mathematics
, specifically in
complex analysis
,
Montel's theorem
is an important result about families of
holomorphic functions
. The
theorem
states that if
F
is a
set
(also called family) of holomorphic functions defined on an
open
and
connected
subset
D
of the
complex numbers
C
, then
F
is a
normal family
if and only if
F
is a
locally bounded set
. The key part of this theorem can be reformulated as follows. Any locally bounded
sequence
of holomorphic functions
f
n
defined on
D
has a
subsequence
which
converges uniformly
on
compact
subsets to a holomorphic function
f
.
References
John B. Conway. (1978)
Functions of One Complex Variable I
. Springer-Verlag, New York, New York.
See also
Arzel-Ascoli theorem
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