<s>
The	O
indefinite	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
space	I-Algorithm
itself	O
is	O
not	O
necessarily	O
a	O
Hilbert	O
space	O
;	O
but	O
the	O
existence	O
of	O
a	O
positive	O
semi-definite	O
inner	O
product	O
on	O
implies	O
that	O
one	O
can	O
form	O
a	O
quotient	O
space	O
on	O
which	O
there	O
is	O
a	O
positive	O
definite	O
inner	O
product	O
.	O
</s>
<s>
Given	O
a	O
strong	O
enough	O
topology	B-Architecture
on	O
this	O
quotient	O
space	O
,	O
it	O
has	O
the	O
structure	O
of	O
a	O
Hilbert	O
space	O
,	O
and	O
many	O
objects	O
of	O
interest	O
in	O
typical	O
applications	O
fall	O
into	O
this	O
quotient	O
space	O
.	O
</s>
<s>
An	O
indefinite	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
space	I-Algorithm
is	O
called	O
a	O
Krein	B-Algorithm
space	I-Algorithm
(	O
or	O
-space	O
)	O
if	O
is	O
positive	O
definite	O
and	O
possesses	O
a	O
majorant	O
topology	B-Architecture
.	O
</s>
<s>
Krein	B-Algorithm
spaces	I-Algorithm
are	O
named	O
in	O
honor	O
of	O
the	O
Soviet	O
mathematician	O
Mark	O
Grigorievich	O
Krein	O
(	O
3	O
April	O
1907	O
–	O
17	O
October	O
1989	O
)	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
Krein	B-Algorithm
spaces	I-Algorithm
it	O
is	O
common	O
to	O
call	O
such	O
an	O
hermitian	O
form	O
an	O
indefinite	O
inner	O
product	O
.	O
</s>
<s>
Let	O
our	O
indefinite	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
space	I-Algorithm
also	O
be	O
equipped	O
with	O
a	O
decomposition	O
into	O
a	O
pair	O
of	O
subspaces	O
,	O
called	O
the	O
fundamental	O
decomposition	O
,	O
which	O
respects	O
the	O
complex	O
structure	O
on	O
.	O
</s>
<s>
Hence	O
the	O
corresponding	O
linear	B-Algorithm
projection	I-Algorithm
operators	I-Algorithm
coincide	O
with	O
the	O
identity	O
on	O
and	O
annihilate	O
,	O
and	O
they	O
commute	O
with	O
multiplication	O
by	O
the	O
of	O
the	O
complex	O
structure	O
.	O
</s>
<s>
If	O
this	O
decomposition	O
is	O
such	O
that	O
and	O
,	O
then	O
is	O
called	O
an	O
indefinite	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
space	I-Algorithm
;	O
if	O
,	O
then	O
is	O
called	O
a	O
Krein	B-Algorithm
space	I-Algorithm
,	O
subject	O
to	O
the	O
existence	O
of	O
a	O
majorant	O
topology	B-Architecture
on	O
(	O
a	O
locally	O
convex	O
topology	B-Architecture
where	O
the	O
inner	O
product	O
is	O
jointly	O
continuous	O
)	O
.	O
</s>
<s>
On	O
a	O
Krein	B-Algorithm
space	I-Algorithm
,	O
the	O
Hilbert	O
inner	O
product	O
is	O
positive	O
definite	O
,	O
giving	O
the	O
structure	O
of	O
a	O
Hilbert	O
space	O
(	O
under	O
a	O
suitable	O
topology	B-Architecture
)	O
.	O
</s>
<s>
Thus	O
is	O
a	O
Hilbert	O
space	O
(	O
given	O
a	O
suitable	O
topology	B-Architecture
)	O
.	O
</s>
<s>
Krein	B-Algorithm
spaces	I-Algorithm
arise	O
naturally	O
in	O
situations	O
where	O
the	O
indefinite	O
inner	O
product	O
has	O
an	O
analytically	O
useful	O
property	O
(	O
such	O
as	O
Lorentz	O
invariance	O
)	O
which	O
the	O
Hilbert	O
inner	O
product	O
lacks	O
.	O
</s>
<s>
All	O
topological	O
notions	O
in	O
a	O
Krein	B-Algorithm
space	I-Algorithm
,	O
like	O
continuity	O
,	O
closed-ness	O
of	O
sets	O
,	O
and	O
the	O
spectrum	O
of	O
an	O
operator	O
on	O
,	O
are	O
understood	O
with	O
respect	O
to	O
this	O
Hilbert	O
space	O
topology	B-Architecture
.	O
</s>
<s>
If	O
,	O
the	O
Krein	B-Algorithm
space	I-Algorithm
is	O
called	O
a	O
Pontryagin	B-Algorithm
space	I-Algorithm
or	O
-space	O
.	O
</s>
<s>
A	O
symmetric	O
operator	O
A	O
on	O
an	O
indefinite	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
space	I-Algorithm
K	O
with	O
domain	O
K	O
is	O
called	O
a	O
Pesonen	B-Algorithm
operator	I-Algorithm
if	O
(	O
x	O
,	O
x	O
)	O
=	O
0	O
=	O
(	O
x	O
,	O
Ax	O
)	O
implies	O
x	O
=	O
0	O
.	O
</s>
