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'' $  G $-
space''

A pair of objects  $  ( E , G ) $, 
the first of which is a vector space  $  E $
over the field of complex numbers, while the second is a bilinear (more precisely, sesquilinear) form  $  G $
on  $  E $; 
this form is also called a  $  G $-
metric. If  $  G $
is a positive-definite (a so-called definite) form, then it is a scalar product in  $  E $, 
and one can use it to canonically introduce (cf., e.g., [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]]) a norm and a distance (i.e. an ordinary metric) for the elements of  $  E $. 
In the case of a general sesquilinear form  $  G $
there is neither a norm nor a metric canonically related to  $  G $, 
and the phrase  "G-metric"  only recalls the close relation of definite sesquilinear forms with certain metrics in vector spaces.

The theory of finite-dimensional spaces with an indefinite metric, more often called bilinear metric spaces, or spaces with a bilinear metric, was developed already by G. Frobenius, and is expounded in courses on linear algebra (cf. [[#References|[1]]]).

The main purpose of the general theory of spaces with an indefinite metric is the separation and study of relatively simple, but for applications important, classes of non-self-adjoint operators in a Hilbert space (cf. [[Non-self-adjoint operator|Non-self-adjoint operator]]). Spaces with an indefinite metric were for the first time introduced by L.S. Pontryagin [[#References|[2]]] (for more detail, see [[Pontryagin space|Pontryagin space]]).

The theory of spaces with an indefinite metric has been developed in two directions — their geometry and linear operations on them.

In the geometry of spaces with an indefinite metric one basically studies: a) the relation between the  $  G $-
metric and various topologies on  $  E $; 
b) the classification of vector subspaces (linear manifolds) in  $  E $
relative to the  $  G $-
metric (especially, the so-called definite subspaces, see below); c) the properties of  $  G $-
projections; and d) bases of  $  G $-
spaces.

In the case of a Hermitian  $  G $-
metric (a  $  G  ^ {H} $-
metric), i.e. such that  $  G ( x , y ) = \overline{ {G ( y , x ) }}\; $
for all  $  x , y \in E $, 
the most important results and concepts in the geometry of spaces with an indefinite metric are as follows. Suppose that each vector  $  y \in E $
is put in correspondence with a linear functional  $  G _ {y} :  x \rightarrow G ( x , y ) $, 
$  x \in E $. 
A topology  $  \tau $
on  $  E $
is called subordinate to the  $  G $-
metric if  $  G _ {y} $
is continuous in  $  \tau $
for all  $  y \in E $; 
$  \tau $
is called compatible with the  $  G $-
metric if it is subordinate to  $  G $
and if every  $  \tau $-
continuous functional has the form  $  G _ {y} $, 
$  y \in E $. 
In a space  $  E $
with an indefinite metric one cannot specify more than one Fréchet topology subordinate to  $  G $, 
and not every  $  G $-
metric allows such a topology (cf. [[#References|[4]]]). If a topology, subordinate to the  $  G $-
metric, is a pre-Hilbert topology on  $  E $
and is given by a scalar product  $  H ( \cdot , \cdot ) $
in  $  E $, 
then  $  H $
is called a Hermitian non-negative majorant of  $  G $; 
in this case

$$ 
| G ( x , y ) |  ^ {2}  \leq   CH ( x , x ) H ( y , y ) ,\ \ 
C = \textrm{ const } ,\ \ 
x , y \in E .
$$

After completing in the  $  H $-
norm one obtains a Hilbert space with indefinite metric  $  ( \widetilde{E}  , \widetilde{G}  ) $, 
where  $  \widetilde{G}  $
is the continuous extension of  $  G $
to the entire space  $  \widetilde{E}  $. 
Here,  $  \widetilde{G}  $
may turn out to be a degenerate metric, even if  $  G $
is non-degenerate. This degeneration does not occur if  $  G $
is a non-degenerate metric and if the largest of the dimensions  $  \kappa $
of the positive subspaces of  $  E $
is finite. In the latter case one obtains the Pontryagin space  $  \Pi _  \kappa  $.

A subspace  $  L $
in a space  $  ( E , G ) $
with an indefinite metric is called a positive subspace, a negative subspace (a more general name is: a definite subspace) or a neutral subspace, depending on whether  $  G ( x , x ) > 0 $, 
$  G ( x , x ) < 0 $
or  $  G ( x , x ) = 0 $
for all  $  x \in L $. 
A subspace is called maximally positive if it is positive and cannot be extended with preservation of this property. Every subspace of the type indicated above is contained in a maximal subspace of the same type.

An important part in the classification of subspaces in spaces with an indefinite metric is played by the notions of a canonical decomposition and a  $  G $-
orthogonal projection.

A vector  $  x \in E $
is called  $  G $-
orthogonal to a subspace  $  L \subset  E $(
is isotropic with respect to  $  L $) 
if  $  G ( x , y ) = 0 $
for all  $  y \in L $. 
A subspace  $  L $
is called degenerate if it contains at least one non-zero vector that is isotropic with respect to  $  L $.

If  $  L $
is a subspace in a space  $  E $
with an indefinite metric, then  $  L  ^  \prime  = \{ {y } : {G ( x , y ) = 0  \textrm{ for  all  }   x \in L } \} $
is its  $  G $-
orthogonal complement. Always  $  L  ^ {\prime\prime} = L  ^  \tau  $, 
where  $  \tau $
is any topology compatible with  $  G $. 
The  $  G $-
orthogonal complement  $  L  ^  \prime  $
of a degenerate vector subspace  $  L $
is a degenerate vector subspace that is closed in a topology  $  \tau $
compatible with  $  G $, 
and  $  L \cap L  ^  \prime  $
is the vector subspace of isotropic elements. A subspace  $  L $
is called projection complete if each  $  y \in E $
has a  $  G $-
projection on  $  L $, 
i.e. if there is an  $  y _ {0} \in L $
for which  $  G ( x , y - y _ {0} ) = 0 $
for every  $  x \in L $. 
Uniqueness of a  $  G $-
projection on  $  L $
is equivalent with  $  L $
being a non-degenerate subspace, while its existence depends on the continuity of the functional  $  G _ {y} $
in topologies on  $  L $
compatible with  $  G $. 
If  $  N $
and  $  M $
are  $  G $-
orthogonal subspaces and  $  M + N = E $, 
then  $  M $
and  $  N $
are projection complete; if  $  L $
is a projection-complete subspace, then  $  L + L  ^  \prime  = E $; 
the sum is the direct sum if  $  E $
is a non-degenerate space with an indefinite metric.

Suppose that  $  L $
is a definite subspace in a space with an indefinite metric  $  E $. 
It is called regular if every functional  $  G _ {y} $, 
$  y \in E $, 
is continuous on  $  E $
in the norm  $  \| x \| _ {G} = | G ( x , x ) | ^ {1/2 } $. 
Otherwise it is called singular. Every non-degenerate infinite-dimensional space with an indefinite metric contains singular subspaces. A definite subspace  $  L $
is projection complete if and only if it is regular and if for every  $  y \in E $
there is an  $  x \in L $
such that

$$ 
\| x \| _ {G}  ^ {2}  =  \| G _ {y} \| _ {G}  ^ {2}  =  G ( x , y ) .
$$

Linear operators in spaces with an indefinite metric have been studied mainly in Hilbert spaces with an indefinite metric; for Banach analogues there is a survey in [[#References|[8]]].

As in the case of Hilbert spaces with an indefinite metric, an important tool in the study of the geometry of spaces with an indefinite metric and of linear operators in spaces  $  ( E , G ) $
endowed with some topology compatible with  $  G $, 
are the so-called  $  G $-
orthogonal bases in  $  E $, 
i.e. bases  $  \{ e _ {n} \} $
of the topological vector space  $  E $
for which  $  G( e _ {k} , e _ {n} ) = \pm  \delta _ {kn} $, 
$  k , n = 1 , 2 , .  .  . $(
cf. [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Foundations of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Hermitian operators in spaces with indefinite metric"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''8'''  (1944)  pp. 243–280  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Iokhvidov,   M.G. Krein,   "Spectral theory in spaces with an indefinite metric I"  ''Transl. Amer. Math. Soc.'' , '''13'''  (1960)  pp. 105–176  ''Trudy Moskov. Mat. Obshch.'' , '''5'''  (1956)  pp. 367–432</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.P. Ginzburg,   I.S. Iokhvidov,   "The geometry of infinite-dimensional spaces with a bilinear metric"  ''Russian Math. Surveys'' , '''17''' :  4  (1962)  pp. 1–51  ''Uspekhi Mat. Nauk'' , '''17''' :  4  (1962)  pp. 3–56</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.G. Krein,   "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230140.png" />-spaces and the theory of operators in these spaces" , ''Second Math. Summer School'' , '''1''' , Kiev  (1965)  pp. 15–92  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T.Ya. Azizov,   I.S. Iokhvidov,   "Linear operators in Hilbert spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230141.png" />-metric"  ''Russian Math. Surveys'' , '''26''' :  4  (1971)  pp. 45–97  ''Uspekhi Mat. Nauk'' , '''26''' :  4  (1971)  pp. 43–92</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K.L. Nagy,   "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.S. Iokhvidov,   "Banach spaces with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230142.png" />-metric and certain classes of linear operators in these spaces"  ''Izv. Akad. Nauk MoldavSSR'' , '''1'''  (1968)  pp. 60–80  (In Russian)</TD></TR></table>

====Comments====
A topology  $  \tau $
that is compatible with the  $  G $-
metric is also called an admissible topology. For an admissible topology,  $  L  ^  \tau  $
denotes the  $  \tau $-
closure of  $  L $
in  $  E $. 
The  $  G $-
orthogonal complement is called the orthogonal companion in [[#References|[a2]]]. The weak topology on the  $  G $-
space  $  E $
is the locally convex topology defined by the family of semi-norms (cf. [[Semi-norm|Semi-norm]])

$$ 
p _ {y} ( x)  =  | G ( x , y) | .
$$

It is an admissible topology, and the weakest such. As a consequence of the double orthogonal complement theorem,  $  L  ^ {\prime\prime} = L  ^  \tau  $, 
one thus has that  $  L  ^ {\prime\prime} = L $
if and only if  $  L $
is weakly closed.

For additional information about spaces with an indefinite metric see [[Krein space|Krein space]] and [[#References|[a1]]]–[[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya. Azizov,   I.S. [I.S. Iokhvidov] Iohidov,   "Linear operators in spaces with an indefinite metric" , Wiley  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bognár,   "Indefinite inner product spaces" , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. [I. Gokhberg] Gohberg,   P. Lancaster,   L. Rodman,   "Matrices and indefinite scalar products" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.S. [I.S. Iokhvidov] Iohidov,   M.G. Krein,   H. Langer,   "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag  (1982)</TD></TR></table>