<s>
In	O
mathematics	O
,	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
is	O
a	O
transformation	B-Algorithm
that	O
preserves	O
the	O
inner	O
product	O
:	O
the	O
inner	O
product	O
of	O
two	O
vectors	O
before	O
the	O
transformation	B-Algorithm
is	O
equal	O
to	O
their	O
inner	O
product	O
after	O
the	O
transformation	B-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
is	O
an	O
isomorphism	O
between	O
two	O
inner	O
product	O
spaces	O
(	O
such	O
as	O
Hilbert	O
spaces	O
)	O
.	O
</s>
<s>
A	O
unitary	B-Algorithm
transformation	I-Algorithm
is	O
an	O
isometry	O
,	O
as	O
one	O
can	O
see	O
by	O
setting	O
in	O
this	O
formula	O
.	O
</s>
<s>
In	O
the	O
case	O
when	O
and	O
are	O
the	O
same	O
space	O
,	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
is	O
an	O
automorphism	O
of	O
that	O
Hilbert	O
space	O
,	O
and	O
then	O
it	O
is	O
also	O
called	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
