View source for Space with an indefinite metric

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862302.png" />-space''

A pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862303.png" />, the first of which is a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862304.png" /> over the field of complex numbers, while the second is a bilinear (more precisely, sesquilinear) form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862305.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862306.png" />; this form is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862308.png" />-metric. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862309.png" /> is a positive-definite (a so-called definite) form, then it is a scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623010.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623011.png" />. In the case of a general sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623012.png" /> there is neither a norm nor a metric canonically related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623013.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623014.png" />-metric and various topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623015.png" />; b) the classification of vector subspaces (linear manifolds) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623016.png" /> relative to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623017.png" />-metric (especially, the so-called definite subspaces, see below); c) the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623018.png" />-projections; and d) bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623019.png" />-spaces.

In the case of a Hermitian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623021.png" />-metric (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623023.png" />-metric), i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623025.png" />, the most important results and concepts in the geometry of spaces with an indefinite metric are as follows. Suppose that each vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623026.png" /> is put in correspondence with a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623028.png" />. A topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623032.png" /> is called subordinate to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623034.png" />-metric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623035.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623037.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623038.png" /> is called compatible with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623040.png" />-metric if it is subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623041.png" /> and if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623042.png" />-continuous functional has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623044.png" />. In a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623045.png" /> with an indefinite metric one cannot specify more than one Fréchet topology subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623046.png" />, and not every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623047.png" />-metric allows such a topology (cf. [[#References|[4]]]). If a topology, subordinate to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623048.png" />-metric, is a pre-Hilbert topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623049.png" /> and is given by a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623052.png" /> is called a Hermitian non-negative majorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623054.png" />; in this case

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623055.png" /></td> </tr></table>

After completing in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623056.png" />-norm one obtains a Hilbert space with indefinite metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623058.png" /> is the continuous extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623059.png" /> to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623060.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623061.png" /> may turn out to be a degenerate metric, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623062.png" /> is non-degenerate. This degeneration does not occur if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623063.png" /> is a non-degenerate metric and if the largest of the dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623064.png" /> of the positive subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623065.png" /> is finite. In the latter case one obtains the Pontryagin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623066.png" />.

A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623067.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623068.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623070.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623071.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623072.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623074.png" />-orthogonal projection.

A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623075.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623077.png" />-orthogonal to a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623078.png" /> (is isotropic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623079.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623081.png" />. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623082.png" /> is called degenerate if it contains at least one non-zero vector that is isotropic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623083.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623084.png" /> is a subspace in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623085.png" /> with an indefinite metric, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623086.png" /> is its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623088.png" />-orthogonal complement. Always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623089.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623090.png" /> is any topology compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623091.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623092.png" />-orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623093.png" /> of a degenerate vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623094.png" /> is a degenerate vector subspace that is closed in a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623095.png" /> compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623096.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623097.png" /> is the vector subspace of isotropic elements. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623098.png" /> is called projection complete if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623099.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230101.png" />-projection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230102.png" />, i.e. if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230103.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230104.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230105.png" />. Uniqueness of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230106.png" />-projection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230107.png" /> is equivalent with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230108.png" /> being a non-degenerate subspace, while its existence depends on the continuity of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230109.png" /> in topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230110.png" /> compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230111.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230113.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230114.png" />-orthogonal subspaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230115.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230117.png" /> are projection complete; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230118.png" /> is a projection-complete subspace, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230119.png" />; the sum is the direct sum if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230120.png" /> is a non-degenerate space with an indefinite metric.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230121.png" /> is a definite subspace in a space with an indefinite metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230122.png" />. It is called regular if every functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230124.png" />, is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230125.png" /> in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230126.png" />. Otherwise it is called singular. Every non-degenerate infinite-dimensional space with an indefinite metric contains singular subspaces. A definite subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230127.png" /> is projection complete if and only if it is regular and if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230128.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230129.png" /> such that

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230131.png" /> endowed with some topology compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230132.png" />, are the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230134.png" />-orthogonal bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230135.png" />, i.e. bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230136.png" /> of the topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230137.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230139.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230143.png" /> that is compatible with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230144.png" />-metric is also called an admissible topology. For an admissible topology, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230146.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230147.png" />-closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230148.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230149.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230150.png" />-orthogonal complement is called the orthogonal companion in [[#References|[a2]]]. The weak topology on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230152.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230153.png" /> is the locally convex topology defined by the family of semi-norms (cf. [[Semi-norm|Semi-norm]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230154.png" /></td> </tr></table>

It is an admissible topology, and the weakest such. As a consequence of the double orthogonal complement theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230155.png" />, one thus has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230156.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s086230157.png" /> 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>