𝐺_{2𝑛} spaces
Authors: Koehler Donald O. 
Published: June 2010

A complex normed linear space

X
X

will be called a

G

2
n

{G_{2n}}

space if and only if there is a mapping

⟨

⋅
,
…
,
⋅

⟩

\left \langle { \cdot , \ldots , \cdot } \right \rangle

from

X

2
n

{X^{2n}}

into the complex numbers such that:

x
k

→

⟨

x
1

,
…
,

x

2
n

⟩

{x_k} \to \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle

is linear for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 1 comma ellipsis comma n semicolon mathematical left-angle x 1 comma ellipsis comma x Subscript 2 n Baseline mathematical right-angle equals mathematical left-angle x Subscript 2 n Baseline comma ellipsis comma x 1 mathematical right-angle Superscript minus">

k
=
1
,
…
,
n
;

⟨

x
1

,
…
,

x

2
n

⟩

=

⟨

x

2
n

,
…
,

x
1

⟩

−

k = 1, \ldots ,n;\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle = {\left \langle {{x_{2n}}, \ldots ,{x_1}} \right \rangle ^ - }

; and

⟨

x
,
…
,
x

⟩

1

/

2
n

=

|

|

x

|

|

{\left \langle {x, \ldots ,x} \right \rangle ^{1/2n}} = ||x||

. The basic models are the

L

2
n

{L^{2n}}

spaces, but one also has that every inner product space is a

G

2
n

{G_{2n}}

space for every integer

n
n

. Hence

G

2
n

{G_{2n}}

spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a

G

2
n

{G_{2n}}

space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map

U
U

from

X
X

to

X
X

is an isometry if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper U left-parenthesis x 1 right-parenthesis comma ellipsis comma upper U left-parenthesis x Subscript 2 n Baseline right-parenthesis mathematical right-angle equals mathematical left-angle x 1 comma ellipsis comma x Subscript 2 n Baseline mathematical right-angle">

⟨

U
(

x
1

)
,
…
,
U
(

x

2
n

)

⟩

=

⟨

x
1

,
…
,

x

2
n

⟩

\left \langle {U({x_1}), \ldots ,U({x_{2n}})} \right \rangle = \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle

for all

x
1

,
…
,

x

2
n

{x_1}, \ldots ,{x_{2n}}

. This then provides a way to construct all of the isometries of a finite dimensional

G

2
n

{G_{2n}}

space. Of particular interest are the

CBS
⁡

G

2
n

\operatorname {CBS} {G_{2n}}

spaces in which
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue mathematical left-angle x 1 comma ellipsis comma x Subscript 2 n Baseline mathematical right-angle EndAbsoluteValue less-than-over-equals StartAbsoluteValue EndAbsoluteValue x 1 StartAbsoluteValue EndAbsoluteValue midline-horizontal-ellipsis StartAbsoluteValue EndAbsoluteValue x Subscript 2 n Baseline StartAbsoluteValue EndAbsoluteValue">

|

⟨

x
1

,
…
,

x

2
n

⟩

|

≦

|

|

x
1

|

|

⋯

|

|

x

2
n

|

|

|\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle | \leqq ||{x_1}|| \cdots ||{x_{2n}}||

. These spaces have many properties similar to inner product spaces. An operator

A
A

on a complete

CBS
⁡

G

2
n

\operatorname {CBS} {G_{2n}}

space is said to be symmetric if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle x 1 comma ellipsis comma upper A left-parenthesis x Subscript i Baseline right-parenthesis comma ellipsis comma x Subscript 2 n Baseline mathematical right-angle equals mathematical left-angle x 1 comma ellipsis comma upper A left-parenthesis x Subscript j Baseline right-parenthesis comma ellipsis comma x Subscript 2 n Baseline mathematical right-angle">

⟨

x
1

,
…
,
A
(

x
i

)
,
…
,

x

2
n

⟩

=

⟨

x
1

,
…
,
A
(

x
j

)
,
…
,

x

2
n

⟩

\left \langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right \rangle = \left \langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right \rangle

for all

i
i

and

j
j

. It is easy to show that these operators are scalar and that on

L

2
n

,
n
>
1

{L^{2n}},n > 1

, they characterize multiplication by a real

L
∞

{L^\infty }

function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial

G

2
n

{G_{2n}}

spaces.