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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s0862302.png" />-space''
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$#C+1 = 138 : ~/encyclopedia/old_files/data/S086/S.0806230 Space with an indefinite metric,
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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.
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'' $  G $-
 +
space''
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 +
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 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]]]).
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The theory of spaces with an indefinite metric has been developed in two directions — their geometry and linear operations on them.
 
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 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 <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
+
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
  
<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>
+
$$
 +
| G ( x , y ) |  ^ {2}  \leq  CH ( x , x ) H ( y , y ) ,\ \
 +
C = \textrm{ const } ,\ \
 +
x , y \in E .
 +
$$
  
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" />.
+
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 <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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086230/s08623074.png" />-orthogonal projection.
+
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 <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" />.
+
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 <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.
+
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 <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
+
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
  
<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/s086230130.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {G}  ^ {2}  = \| G _ {y} \| _ {G}  ^ {2}  = G ( x , y ) .
 +
$$
  
 
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]]].
 
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]]]).
+
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====
 
====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>
 
<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====
 
====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]])
+
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]])
  
<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>
+
$$
 +
p _ {y} ( x)  = | G ( x , y) | .
 +
$$
  
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.
+
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]]].
 
For additional information about spaces with an indefinite metric see [[Krein space|Krein space]] and [[#References|[a1]]]–[[#References|[a4]]].

Latest revision as of 08:22, 6 June 2020


$ 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. [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). Spaces with an indefinite metric were for the first time introduced by L.S. Pontryagin [2] (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. [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 [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. [4]).

References

[1] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)
[2] L.S. Pontryagin, "Hermitian operators in spaces with indefinite metric" Izv. Akad. Nauk SSSR Ser. Mat. , 8 (1944) pp. 243–280 (In Russian)
[3] 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
[4] 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
[5] M.G. Krein, "Introduction to the geometry of indefinite -spaces and the theory of operators in these spaces" , Second Math. Summer School , 1 , Kiev (1965) pp. 15–92 (In Russian)
[6] T.Ya. Azizov, I.S. Iokhvidov, "Linear operators in Hilbert spaces with -metric" Russian Math. Surveys , 26 : 4 (1971) pp. 45–97 Uspekhi Mat. Nauk , 26 : 4 (1971) pp. 43–92
[7] K.L. Nagy, "State vector spaces with indefinite metric in quantum field theory" , Noordhoff (1966)
[8] I.S. Iokhvidov, "Banach spaces with a -metric and certain classes of linear operators in these spaces" Izv. Akad. Nauk MoldavSSR , 1 (1968) pp. 60–80 (In Russian)

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 [a2]. The weak topology on the $ G $- space $ E $ is the locally convex topology defined by the family of semi-norms (cf. 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 and [a1][a4].

References

[a1] T.Ya. Azizov, I.S. [I.S. Iokhvidov] Iohidov, "Linear operators in spaces with an indefinite metric" , Wiley (1989) (Translated from Russian)
[a2] J. Bognár, "Indefinite inner product spaces" , Springer (1974)
[a3] I. [I. Gokhberg] Gohberg, P. Lancaster, L. Rodman, "Matrices and indefinite scalar products" , Birkhäuser (1983)
[a4] 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)
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