<s>
In	O
the	O
mathematical	O
study	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
the	O
Bergman	B-Algorithm
kernel	I-Algorithm
,	O
named	O
after	O
Stefan	O
Bergman	O
,	O
is	O
the	O
reproducing	O
kernel	O
for	O
the	O
Hilbert	O
space	O
(	O
RKHS	O
)	O
of	O
all	O
square	B-Algorithm
integrable	I-Algorithm
holomorphic	O
functions	O
on	O
a	O
domain	O
D	O
inCn	O
.	O
</s>
<s>
In	O
detail	O
,	O
let	O
L2(D )	O
be	O
the	O
Hilbert	O
space	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
on	O
D	O
,	O
and	O
let	O
L2	O
,	O
h(D )	O
denote	O
the	O
subspace	O
consisting	O
of	O
holomorphic	O
functions	O
in	O
L2(D )	O
:	O
that	O
is	O
,	O
</s>
<s>
for	O
every	O
compact	O
subset	O
K	O
of	O
D	O
.	O
Thus	O
convergence	O
of	O
a	O
sequence	O
of	O
holomorphic	O
functions	O
in	O
L2(D )	O
implies	O
also	O
compact	B-Algorithm
convergence	I-Algorithm
,	O
and	O
so	O
the	O
limit	O
function	O
is	O
also	O
holomorphic	O
.	O
</s>
<s>
Since	O
the	O
inner	O
product	O
on	O
this	O
space	O
is	O
manifestly	O
invariant	O
under	O
biholomorphisms	O
of	O
D	O
,	O
the	O
Bergman	B-Algorithm
kernel	I-Algorithm
and	O
the	O
associated	O
Bergman	O
metric	O
are	O
therefore	O
automatically	O
invariant	O
under	O
the	O
automorphism	O
group	O
of	O
the	O
domain	O
.	O
</s>
