<s>
This	O
article	O
describes	O
various	O
modes	O
(	O
senses	O
or	O
species	O
)	O
of	O
convergence	B-Algorithm
in	O
the	O
settings	O
where	O
they	O
are	O
defined	O
.	O
</s>
<s>
Convergence	B-Algorithm
can	O
be	O
defined	O
in	O
terms	O
of	O
sequences	B-Algorithm
in	O
first-countable	O
spaces	O
.	O
</s>
<s>
Nets	O
are	O
a	O
generalization	O
of	O
sequences	B-Algorithm
that	O
are	O
useful	O
in	O
spaces	O
which	O
are	O
not	O
first	O
countable	O
.	O
</s>
<s>
Filters	O
further	O
generalize	O
the	O
concept	O
of	O
convergence	B-Algorithm
.	O
</s>
<s>
In	O
metric	O
spaces	O
,	O
one	O
can	O
define	O
Cauchy	O
sequences	B-Algorithm
.	O
</s>
<s>
Convergence	B-Algorithm
implies	O
"	O
Cauchy-convergence	O
"	O
,	O
and	O
Cauchy-convergence	O
,	O
together	O
with	O
the	O
existence	O
of	O
a	O
convergent	O
subsequence	O
implies	O
convergence	B-Algorithm
.	O
</s>
<s>
The	O
concept	O
of	O
completeness	O
of	O
metric	O
spaces	O
,	O
and	O
its	O
generalizations	O
is	O
defined	O
in	O
terms	O
of	O
Cauchy	O
sequences	B-Algorithm
.	O
</s>
<s>
In	O
a	O
topological	O
abelian	O
group	O
,	O
convergence	B-Algorithm
of	O
a	O
series	O
is	O
defined	O
as	O
convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
partial	O
sums	O
.	O
</s>
<s>
An	O
important	O
concept	O
when	O
considering	O
series	O
is	O
unconditional	O
convergence	B-Algorithm
,	O
which	O
guarantees	O
that	O
the	O
limit	O
of	O
the	O
series	O
is	O
invariant	O
under	O
permutations	O
of	O
the	O
summands	O
.	O
</s>
<s>
In	O
a	O
normed	O
vector	O
space	O
,	O
one	O
can	O
define	O
absolute	O
convergence	B-Algorithm
as	O
convergence	B-Algorithm
of	O
the	O
series	O
of	O
norms	O
(	O
)	O
.	O
</s>
<s>
Absolute	O
convergence	B-Algorithm
implies	O
Cauchy	O
convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
partial	O
sums	O
(	O
by	O
the	O
triangle	O
inequality	O
)	O
,	O
which	O
in	O
turn	O
implies	O
absolute-convergence	O
of	O
some	O
grouping	O
(	O
not	O
reordering	O
)	O
.	O
</s>
<s>
The	O
norm	O
convergence	B-Algorithm
of	O
absolutely	O
convergent	O
series	O
is	O
an	O
equivalent	O
condition	O
for	O
a	O
normed	O
linear	O
space	O
to	O
be	O
Banach	O
(	O
i.e.	O
</s>
<s>
Absolute	O
convergence	B-Algorithm
and	O
convergence	B-Algorithm
together	O
imply	O
unconditional	O
convergence	B-Algorithm
,	O
but	O
unconditional	O
convergence	B-Algorithm
does	O
not	O
imply	O
absolute	O
convergence	B-Algorithm
in	O
general	O
,	O
even	O
if	O
the	O
space	O
is	O
Banach	O
,	O
although	O
the	O
implication	O
holds	O
in	O
.	O
</s>
<s>
The	O
most	O
basic	O
type	O
of	O
convergence	B-Algorithm
for	O
a	O
sequence	O
of	O
functions	O
(	O
in	O
particular	O
,	O
it	O
does	O
not	O
assume	O
any	O
topological	O
structure	O
on	O
the	O
domain	B-Algorithm
of	O
the	O
functions	O
)	O
is	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
It	O
is	O
defined	O
as	O
convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
values	O
of	O
the	O
functions	O
at	O
every	O
point	O
.	O
</s>
<s>
If	O
the	O
functions	O
take	O
their	O
values	O
in	O
a	O
uniform	O
space	O
,	O
then	O
one	O
can	O
define	O
pointwise	O
Cauchy	O
convergence	B-Algorithm
,	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
and	O
uniform	B-Algorithm
Cauchy	I-Algorithm
convergence	I-Algorithm
of	O
the	O
sequence	O
.	O
</s>
<s>
Pointwise	B-Algorithm
convergence	I-Algorithm
implies	O
pointwise	O
Cauchy-convergence	O
,	O
and	O
the	O
converse	O
holds	O
if	O
the	O
space	O
in	O
which	O
the	O
functions	O
take	O
their	O
values	O
is	O
complete	O
.	O
</s>
<s>
Uniform	B-Algorithm
convergence	I-Algorithm
implies	O
pointwise	B-Algorithm
convergence	I-Algorithm
and	O
uniform	B-Algorithm
Cauchy	I-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Uniform	B-Algorithm
Cauchy	I-Algorithm
convergence	I-Algorithm
and	O
pointwise	B-Algorithm
convergence	I-Algorithm
of	O
a	O
subsequence	O
imply	O
uniform	B-Algorithm
convergence	I-Algorithm
of	O
the	O
sequence	O
,	O
and	O
if	O
the	O
codomain	O
is	O
complete	O
,	O
then	O
uniform	B-Algorithm
Cauchy	I-Algorithm
convergence	I-Algorithm
implies	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
If	O
the	O
domain	B-Algorithm
of	O
the	O
functions	O
is	O
a	O
topological	O
space	O
,	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
(	O
i.e.	O
</s>
<s>
uniform	B-Algorithm
convergence	I-Algorithm
on	O
a	O
neighborhood	O
of	O
each	O
point	O
)	O
and	O
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
(	O
i.e.	O
</s>
<s>
uniform	B-Algorithm
convergence	I-Algorithm
on	O
all	O
compact	O
subsets	O
)	O
may	O
be	O
defined	O
.	O
</s>
<s>
Note	O
that	O
"	O
compact	B-Algorithm
convergence	I-Algorithm
"	O
is	O
always	O
short	O
for	O
"	O
compact	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
"	O
since	O
"	O
compact	O
pointwise	B-Algorithm
convergence	I-Algorithm
"	O
would	O
mean	O
the	O
same	O
thing	O
as	O
"	O
pointwise	B-Algorithm
convergence	I-Algorithm
"	O
(	O
points	O
are	O
always	O
compact	O
)	O
.	O
</s>
<s>
Uniform	B-Algorithm
convergence	I-Algorithm
implies	O
both	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
and	O
compact	B-Algorithm
convergence	I-Algorithm
,	O
since	O
both	O
are	O
local	O
notions	O
while	O
uniform	B-Algorithm
convergence	I-Algorithm
is	O
global	O
.	O
</s>
<s>
If	O
X	O
is	O
locally	O
compact	O
(	O
even	O
in	O
the	O
weakest	O
sense	O
:	O
every	O
point	O
has	O
compact	O
neighborhood	O
)	O
,	O
then	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
is	O
equivalent	O
to	O
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
.	O
</s>
<s>
Pointwise	O
and	O
uniform	B-Algorithm
convergence	I-Algorithm
of	O
series	O
of	O
functions	O
are	O
defined	O
in	O
terms	O
of	O
convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
partial	O
sums	O
.	O
</s>
<s>
For	O
functions	O
taking	O
values	O
in	O
a	O
normed	O
linear	O
space	O
,	O
absolute	O
convergence	B-Algorithm
refers	O
to	O
convergence	B-Algorithm
of	O
the	O
series	O
of	O
positive	O
,	O
real-valued	O
functions	O
.	O
</s>
<s>
"	O
Pointwise	O
absolute	O
convergence	B-Algorithm
"	O
is	O
then	O
simply	O
pointwise	B-Algorithm
convergence	I-Algorithm
of	O
.	O
</s>
<s>
Normal	B-Algorithm
convergence	I-Algorithm
is	O
convergence	B-Algorithm
of	O
the	O
series	O
of	O
non-negative	O
real	O
numbers	O
obtained	O
by	O
taking	O
the	O
uniform	O
(	O
i.e.	O
</s>
<s>
"	O
sup	O
"	O
)	O
norm	O
of	O
each	O
function	O
in	O
the	O
series	O
(	O
uniform	B-Algorithm
convergence	I-Algorithm
of	O
)	O
.	O
</s>
<s>
In	O
Banach	O
spaces	O
,	O
pointwise	O
absolute	O
convergence	B-Algorithm
implies	O
pointwise	B-Algorithm
convergence	I-Algorithm
,	O
and	O
normal	B-Algorithm
convergence	I-Algorithm
implies	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
For	O
functions	O
defined	O
on	O
a	O
topological	O
space	O
,	O
one	O
can	O
define	O
(	O
as	O
above	O
)	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
and	O
compact	O
(	O
uniform	O
)	O
convergence	B-Algorithm
in	O
terms	O
of	O
the	O
partial	O
sums	O
of	O
the	O
series	O
.	O
</s>
<s>
If	O
,	O
in	O
addition	O
,	O
the	O
functions	O
take	O
values	O
in	O
a	O
normed	O
linear	O
space	O
,	O
then	O
local	O
normal	B-Algorithm
convergence	I-Algorithm
(	O
local	O
,	O
uniform	O
,	O
absolute	O
convergence	B-Algorithm
)	O
and	O
compact	O
normal	B-Algorithm
convergence	I-Algorithm
(	O
absolute	O
convergence	B-Algorithm
on	O
compact	O
sets	O
)	O
can	O
be	O
defined	O
.	O
</s>
<s>
Normal	B-Algorithm
convergence	I-Algorithm
implies	O
both	O
local	O
normal	B-Algorithm
convergence	I-Algorithm
and	O
compact	O
normal	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
And	O
if	O
the	O
domain	B-Algorithm
is	O
locally	O
compact	O
(	O
even	O
in	O
the	O
weakest	O
sense	O
)	O
,	O
then	O
local	O
normal	B-Algorithm
convergence	I-Algorithm
implies	O
compact	O
normal	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
If	O
one	O
considers	O
sequences	B-Algorithm
of	O
measurable	O
functions	O
,	O
then	O
several	O
modes	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
that	O
depend	O
on	O
measure-theoretic	O
,	O
rather	O
than	O
solely	O
topological	O
properties	O
,	O
arise	O
.	O
</s>
<s>
This	O
includes	O
pointwise	B-Algorithm
convergence	I-Algorithm
almost-everywhere	O
,	O
convergence	B-Algorithm
in	O
p-mean	O
and	O
convergence	B-Algorithm
in	O
measure	O
.	O
</s>
