# Space with an indefinite metric

$G$- 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) 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. ).

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). Spaces with an indefinite metric were for the first time introduced by L.S. Pontryagin  (for more detail, see 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. ). 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 .

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. ).

How to Cite This Entry:
Space with an indefinite metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Space_with_an_indefinite_metric&oldid=48753
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article