The analytic properties of 𝐺_{2𝑛} spaces
Authors: Koehler Donald O. 
Published: June 2010

A complex vector space X will be called an

F

2
n

{F_{2n}}

space if and only if there is a mapping

⟨
⋅
,
⋯
,
⋅
⟩

\langle \cdot , \cdots , \cdot \rangle

from

X

2
n

{X^{2n}}

into the complex numbers such that:

⟨
x
,
⋯
,
x
⟩
>
0

\langle x, \cdots ,x\rangle > 0

if

x
≠
0
;

x
k

→
⟨

x
1

,
⋯
,

x

2
n

⟩

x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle

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

k
=
1
,
⋯
,
n
;
⟨

x
1

,
⋯
,

x

2
n

⟩
=

⟨

x

2
n

,
⋯
,

x
1

⟩
−

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

where denotes complex conjugate;
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle x Subscript sigma left-parenthesis 1 right-parenthesis Baseline comma midline-horizontal-ellipsis comma x Subscript sigma left-parenthesis n right-parenthesis Baseline comma y Subscript tau left-parenthesis 1 right-parenthesis Baseline comma midline-horizontal-ellipsis comma y Subscript tau left-parenthesis n right-parenthesis Baseline mathematical right-angle equals mathematical left-angle x 1 comma midline-horizontal-ellipsis comma x Subscript n Baseline comma y 1 comma midline-horizontal-ellipsis comma y Subscript n Baseline mathematical right-angle">

⟨

x

σ
(
1
)

,
⋯
,

x

σ
(
n
)

,

y

τ
(
1
)

,
⋯
,

y

τ
(
n
)

⟩
=
⟨

x
1

,
⋯
,

x
n

,

y
1

,
⋯
,

y
n

⟩

\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle

for all permutations

σ
,
τ

\sigma ,\tau

of

{
1
,
⋯
,
n
}

\{ 1, \cdots ,n\}

. In the case of a real vector space the mapping is assumed to be into the reals such that:

⟨
x
,
⋯
,
x
⟩
>
0

\langle x, \cdots ,x\rangle > 0

if

x
≠
0
;

x
k

→
⟨

x
1

,
⋯
,

x

2
n

⟩

x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle

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

k
=
1
,
⋯
,
2
n
;
⟨

x

σ
(
1
)

,
⋯
,

x

σ
(
2
n
)

⟩
=
⟨

x
1

,
⋯
,

x

2
n

⟩

k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle

for all permutations

σ
\sigma

of

{
1
,
⋯
,
2
n
}

\{ 1, \cdots ,2n\}

. In either case, if

‖
x
‖

=

⟨
x
,
⋯
,
x

⟩

1

/

2
n

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

defines a norm, X is called a

G

2
n

{G_{2n}}

space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an

F

2
n

{F_{2n}}

space is a

G

2
n

{G_{2n}}

space if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue mathematical left-angle x comma y comma midline-horizontal-ellipsis comma y mathematical right-angle EndAbsoluteValue Superscript 2 n Baseline less-than-over-equals mathematical left-angle x comma midline-horizontal-ellipsis comma x mathematical right-angle mathematical left-angle y comma midline-horizontal-ellipsis comma y mathematical right-angle Superscript 2 n minus 1">

|

⟨
x
,
y
,
⋯
,
y
⟩

|

2
n

≦
⟨
x
,
⋯
,
x
⟩

⟨
y
,
⋯
,
y

⟩

2
n
−
1

|\langle x,y, \cdots ,y\rangle {|^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}}

and that

G

2
n

{G_{2n}}

spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).