<s>
In	O
mathematics	O
,	O
one	O
normed	O
vector	O
space	O
is	O
said	O
to	O
be	O
continuously	B-Algorithm
embedded	I-Algorithm
in	O
another	O
normed	O
vector	O
space	O
if	O
the	O
inclusion	B-Algorithm
function	I-Algorithm
between	O
them	O
is	O
continuous	O
.	O
</s>
<s>
Several	O
of	O
the	O
Sobolev	O
embedding	O
theorems	O
are	O
continuous	B-Algorithm
embedding	I-Algorithm
theorems	O
.	O
</s>
<s>
for	O
every	O
x	O
in	O
X	O
,	O
then	O
X	O
is	O
said	O
to	O
be	O
continuously	B-Algorithm
embedded	I-Algorithm
in	O
Y	O
.	O
</s>
<s>
Some	O
authors	O
use	O
the	O
hooked	O
arrow	O
"	O
↪	O
"	O
to	O
denote	O
a	O
continuous	B-Algorithm
embedding	I-Algorithm
,	O
i.e.	O
</s>
<s>
"	O
X	O
↪	O
Y	O
"	O
means	O
"	O
X	O
and	O
Y	O
are	O
normed	O
spaces	O
with	O
X	O
continuously	B-Algorithm
embedded	I-Algorithm
in	O
Y	O
"	O
.	O
</s>
<s>
This	O
is	O
a	O
consistent	O
use	O
of	O
notation	O
from	O
the	O
point	O
of	O
view	O
of	O
the	O
category	B-Algorithm
of	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
,	O
in	O
which	O
the	O
morphisms	O
(	O
"	O
arrows	O
"	O
)	O
are	O
the	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
A	O
finite-dimensional	O
example	O
of	O
a	O
continuous	B-Algorithm
embedding	I-Algorithm
is	O
given	O
by	O
a	O
natural	O
embedding	O
of	O
the	O
real	O
line	O
X	O
=	O
R	O
into	O
the	O
plane	O
Y	O
=	O
R2	O
,	O
where	O
both	O
spaces	O
are	O
given	O
the	O
Euclidean	O
norm	O
:	O
</s>
<s>
An	O
infinite-dimensional	O
example	O
of	O
a	O
continuous	B-Algorithm
embedding	I-Algorithm
is	O
given	O
by	O
the	O
Rellich	O
–	O
Kondrachov	O
theorem	O
:	O
let	O
Ω⊆Rn	O
be	O
an	O
open	O
,	O
bounded	B-Algorithm
,	O
Lipschitz	O
domain	O
,	O
and	O
let	O
1≤pn	O
.	O
</s>
<s>
Then	O
the	O
Sobolev	O
space	O
W1	O
,	O
p( 	O
;	O
R	O
)	O
is	O
continuously	B-Algorithm
embedded	I-Algorithm
in	O
the	O
Lp	O
space	O
Lp∗( 	O
;	O
R	O
)	O
.	O
</s>
<s>
Infinite-dimensional	O
spaces	O
also	O
offer	O
examples	O
of	O
discontinuous	B-Algorithm
embeddings	I-Algorithm
.	O
</s>
