- Bitlis Eren University Journal of Science and Technology
- Vol: 5 Issue: 2
- A Partial Solution To An Open Problem
A Partial Solution To An Open Problem
Authors : Şükran Konca
Pages : 0-0
Doi:10.17678/beujst.41573
View : 6 | Download : 5
Publication Date : 2015-12-28
Article Type : Other
Abstract :Let $\left( {X,\left\| {.,...,.} \right\|} \right)$ be a real $n$-normed space, as introduced by S. Gahler [1] in 1969. The set of all bounded multilinear $n$-functionals on $\left( {X,\left\| {.,...,.} \right\|} \right)$ forms a vector space. A bounded multilinear $n$-functional $F$ is defined by $\left\| F \right\|: = {\rm{sup}}\left\{ {\left| {F\left( {{x_1},...,{x_n}} \right)} \right|:\left\| {{x_1},...,{x_n}} \right\| \le 1} \right\}$. \textbf{\bigskip } This formula defines a norm on $X'$ (the space of all bounded multilinear $n$-functionals on $X$). \textbf{\bigskip } Let $Y: = \left\{ {{y_1},...,{y_n}} \right\}$ in $\ell^{q}$, where $q$ is the dual exponent of $p$. \textbf{\bigskip } Batkunde et al. [2] defined the following multilinear $n$-functional on $\ell^{p}$ where $1 \le p < \infty$: \begin{equation*} {F_Y}\left( {{x_1},...,{x_n}} \right): = \frac{1}{{n!}}\sum\limits_{{j_1}} {...} \sum\limits_{{j_n}} {\left| {\begin{array}{*{20}{c}} {{x_{1{j_1}}}} & \cdots & {{x_{1{j_n}}}} \\ \vdots & \ddots & \vdots \\ {{x_{n{j_1}}}} & \ldots & {{x_{n{j_n}}}} \\ \end{array}} \right|} \left| {\begin{array}{*{20}{c}} {{y_{1{j_1}}}} & \cdots & {{y_{1{j_n}}}} \\ \vdots & \ddots & \vdots \\ {{y_{n{j_1}}}} & \ldots & {{y_{n{j_n}}}} \\ \end{array}} \right| \end{equation*} for ${x_1},...,{x_n} \in \ell^{p}$.\textbf{\bigskip } Regarding the $n$-functional on $\left( {\ell^{p},\left\| {.,...,.} \right\|_p^{}} \right)$, an open problem was given by Batkunde et al. [2]. They want to compute the exact norm of ${F_Y}$, especially for $p \ne 2$. In this paper, we deal with a partial solution to this open problem given in their paper.Keywords :