The analytic properties of $G_{2n}$ spaces

Abstract:

A complex vector space X will be called an ${F_{2n}}$ space if and only if there is a mapping $\langle \cdot , \cdots , \cdot \rangle$ from ${X^{2n}}$ into the complex numbers such that: $\langle x, \cdots ,x\rangle > 0$ if $x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle$ is linear for $k = 1, \cdots ,n;\langle {x_1}, \cdots ,{x_{2n}}\rangle = {\langle {x_{2n}}, \cdots ,{x_1}\rangle ^ - }$ where denotes complex conjugate; $\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle$ for all permutations $\sigma ,\tau$ of $\{ 1, \cdots ,n\}$. In the case of a real vector space the mapping is assumed to be into the reals such that: $\langle x, \cdots ,x\rangle > 0$ if $x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle$ is linear for $k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle$ for all permutations $\sigma$ of $\{ 1, \cdots ,2n\}$. In either case, if $\left \| x \right \| = {\langle x, \cdots ,x\rangle ^{1/2n}}$ defines a norm, X is called a ${G_{2n}}$ space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an ${F_{2n}}$ space is a ${G_{2n}}$ space if and only if $|\langle x,y, \cdots ,y\rangle {|^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}}$ and that ${G_{2n}}$ spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).