<s>
The	O
purpose	O
of	O
this	O
article	O
is	O
to	O
serve	O
as	O
an	O
annotated	O
index	O
of	O
various	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
and	O
their	O
logical	O
relationships	O
.	O
</s>
<s>
For	O
an	O
expository	O
article	O
,	O
see	O
Modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Simple	O
logical	O
relationships	O
between	O
different	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
are	O
indicated	O
(	O
e.g.	O
,	O
if	O
one	O
implies	O
another	O
)	O
,	O
formulaically	O
rather	O
than	O
in	O
prose	O
for	O
quick	O
reference	O
,	O
and	O
indepth	O
descriptions	O
and	O
discussions	O
are	O
reserved	O
for	O
their	O
respective	O
articles	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
list	O
of	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
for	O
:	O
</s>
<s>
Convergence	B-Algorithm
,	O
or	O
"	O
topological	B-Algorithm
convergence	I-Algorithm
"	O
for	O
emphasis	O
(	O
i.e.	O
</s>
<s>
-	O
Cauchy-convergence	O
and	O
convergence	B-Algorithm
of	O
a	O
subsequence	O
together	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
U	O
is	O
called	O
"	O
complete	O
"	O
if	O
Cauchy-convergence	O
(	O
for	O
nets	O
)	O
convergence	B-Algorithm
.	O
</s>
<s>
Note	O
:	O
A	O
sequence	O
exhibiting	O
Cauchy-convergence	O
is	O
called	O
a	O
cauchy	O
sequence	O
to	O
emphasize	O
that	O
it	O
may	O
not	O
be	O
convergent	O
.	O
</s>
<s>
-	O
Unconditional	O
convergence	B-Algorithm
convergence	B-Algorithm
(	O
by	O
definition	O
)	O
.	O
</s>
<s>
-	O
Absolute-convergence	O
Cauchy-convergence	O
absolute-convergence	O
of	O
some	O
grouping1	O
.	O
</s>
<s>
-	O
Therefore	O
:	O
N	O
is	O
Banach	O
(	O
complete	O
)	O
if	O
absolute-convergence	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
Absolute-convergence	O
and	O
convergence	B-Algorithm
together	O
unconditional	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
Unconditional	O
convergence	B-Algorithm
absolute-convergence	O
,	O
even	O
if	O
N	O
is	O
Banach	O
.	O
</s>
<s>
-	O
If	O
N	O
is	O
a	O
Euclidean	O
space	O
,	O
then	O
unconditional	O
convergence	B-Algorithm
absolute-convergence	O
.	O
</s>
<s>
-	O
Uniform	B-Algorithm
convergence	I-Algorithm
both	O
pointwise	B-Algorithm
convergence	I-Algorithm
and	O
uniform	B-Algorithm
Cauchy-convergence	I-Algorithm
.	O
</s>
<s>
-	O
Uniform	B-Algorithm
Cauchy-convergence	I-Algorithm
and	O
pointwise	B-Algorithm
convergence	I-Algorithm
of	O
a	O
subsequence	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
For	O
many	O
"	O
global	O
"	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
,	O
there	O
are	O
corresponding	O
notions	O
of	O
a	O
)	O
"	O
local	O
"	O
and	O
b	O
)	O
"	O
compact	O
"	O
convergence	B-Algorithm
,	O
which	O
are	O
given	O
by	O
requiring	O
convergence	B-Algorithm
to	O
occur	O
a	O
)	O
on	O
some	O
neighborhood	O
of	O
each	O
point	O
,	O
or	O
b	O
)	O
on	O
all	O
compact	O
subsets	O
of	O
X	O
.	O
</s>
<s>
Local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
(	O
i.e.	O
</s>
<s>
Compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
(	O
i.e.	O
</s>
<s>
-	O
"	O
Global	O
"	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
imply	O
the	O
corresponding	O
"	O
local	O
"	O
and	O
"	O
compact	O
"	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Uniform	B-Algorithm
convergence	I-Algorithm
both	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
and	O
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
"	O
Local	O
"	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
tend	O
to	O
imply	O
"	O
compact	O
"	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
.	O
</s>
<s>
Local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
Pointwise	B-Algorithm
convergence	I-Algorithm
almost	O
everywhere	O
convergence	B-Algorithm
.	O
</s>
<s>
-	O
Uniform	B-Algorithm
convergence	I-Algorithm
almost	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
-	O
Almost	O
everywhere	O
convergence	B-Algorithm
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
-	O
Almost	O
uniform	B-Algorithm
convergence	I-Algorithm
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
-	O
Lp	O
convergence	B-Algorithm
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
-	O
Convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
convergence	B-Algorithm
in	O
distribution	O
if	O
is	O
a	O
probability	O
measure	O
and	O
the	O
functions	O
are	O
integrable	O
.	O
</s>
<s>
Generally	O
,	O
replacing	O
"	O
convergence	B-Algorithm
"	O
by	O
"	O
absolute-convergence	O
"	O
means	O
one	O
is	O
referring	O
to	O
convergence	B-Algorithm
of	O
the	O
series	O
of	O
nonnegative	O
functions	O
in	O
place	O
of	O
.	O
</s>
<s>
-	O
Uniform	B-Algorithm
absolute-convergence	I-Algorithm
both	O
local	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
and	O
compact	O
(	O
uniform	O
)	O
absolute-convergence	O
.	O
</s>
<s>
Normal	B-Algorithm
convergence	I-Algorithm
both	O
local	O
normal	B-Algorithm
convergence	I-Algorithm
and	O
compact	O
normal	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
-	O
Local	O
normal	B-Algorithm
convergence	I-Algorithm
local	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
.	O
</s>
<s>
Compact	O
normal	B-Algorithm
convergence	I-Algorithm
compact	O
(	O
uniform	O
)	O
absolute-convergence	O
.	O
</s>
<s>
-	O
Local	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
compact	O
(	O
uniform	O
)	O
absolute-convergence	O
.	O
</s>
<s>
Local	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
compact	O
(	O
uniform	O
)	O
absolute-convergence	O
.	O
</s>
