<s>
In	O
mathematics	O
normal	B-Algorithm
convergence	I-Algorithm
is	O
a	O
type	O
of	O
convergence	B-Algorithm
for	O
series	O
of	O
functions	O
.	O
</s>
<s>
Like	O
absolute-convergence	O
,	O
it	O
has	O
the	O
useful	O
property	O
that	O
it	O
is	O
preserved	O
when	O
the	O
order	O
of	O
summation	O
is	O
changed	O
.	O
</s>
<s>
The	O
concept	O
of	O
normal	B-Algorithm
convergence	I-Algorithm
was	O
first	O
introduced	O
by	O
René	O
Baire	O
in	O
1908	O
in	O
his	O
book	O
Leçons	O
sur	O
les	O
théories	O
générales	O
de	O
l'analyse	O
.	O
</s>
<s>
Normal	B-Algorithm
convergence	I-Algorithm
implies	O
,	O
but	O
should	O
not	O
be	O
confused	O
with	O
,	O
uniform	B-Algorithm
absolute	I-Algorithm
convergence	I-Algorithm
,	O
i.e.	O
</s>
<s>
uniform	O
convergence	B-Algorithm
of	O
the	O
series	O
of	O
nonnegative	O
functions	O
.	O
</s>
<s>
Then	O
the	O
series	O
is	O
uniformly	O
convergent	O
(	O
for	O
any	O
ε	O
take	O
n	O
≥	O
1/ε	O
)	O
,	O
but	O
the	O
series	O
of	O
uniform	O
norms	O
is	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
and	O
thus	O
diverges	O
.	O
</s>
<s>
As	O
well	O
,	O
normal	B-Algorithm
convergence	I-Algorithm
of	O
a	O
series	O
is	O
different	O
from	O
norm-topology	O
convergence	B-Algorithm
,	O
i.e.	O
</s>
<s>
convergence	B-Algorithm
of	O
the	O
partial	O
sum	O
sequence	O
in	O
the	O
topology	O
induced	O
by	O
the	O
uniform	O
norm	O
.	O
</s>
<s>
Normal	B-Algorithm
convergence	I-Algorithm
implies	O
norm-topology	O
convergence	B-Algorithm
if	O
and	O
only	O
if	O
the	O
space	O
of	O
functions	O
under	O
consideration	O
is	O
complete	O
with	O
respect	O
to	O
the	O
uniform	O
norm	O
.	O
</s>
<s>
(	O
The	O
converse	O
does	O
not	O
hold	O
even	O
for	O
complete	O
function	O
spaces	O
:	O
for	O
example	O
,	O
consider	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
as	O
a	O
sequence	O
of	O
constant	O
functions	O
)	O
.	O
</s>
<s>
Note	O
:	O
if	O
X	O
is	O
locally	O
compact	O
(	O
even	O
in	O
the	O
weakest	O
sense	O
)	O
,	O
local	O
normal	B-Algorithm
convergence	I-Algorithm
and	O
compact	O
normal	B-Algorithm
convergence	I-Algorithm
are	O
equivalent	O
.	O
</s>
<s>
Every	O
normal	O
convergent	B-Algorithm
series	I-Algorithm
is	O
uniformly	O
convergent	O
,	O
locally	O
uniformly	O
convergent	O
,	O
and	O
compactly	O
uniformly	O
convergent	O
.	O
</s>
<s>
This	O
is	O
very	O
important	O
,	O
since	O
it	O
assures	O
that	O
any	O
re-arrangement	O
of	O
the	O
series	O
,	O
any	O
derivatives	O
or	O
integrals	O
of	O
the	O
series	O
,	O
and	O
sums	O
and	O
products	O
with	O
other	O
convergent	B-Algorithm
series	I-Algorithm
will	O
converge	O
to	O
the	O
"	O
correct	O
"	O
value	O
.	O
</s>
<s>
That	O
is	O
,	O
for	O
every	O
bijection	B-Algorithm
,	O
is	O
normally	O
convergent	O
to	O
.	O
</s>
