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## A Partial Solution To An Open Problem

#### Şükran KONCA [1]

##### 1611 646

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.

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Journal Section Articles Author: Şükran KONCA Publication Date: December 28, 2015
 Bibtex @ { beuscitech477729, journal = {Bitlis Eren University Journal of Science and Technology}, issn = {}, eissn = {2146-7706}, address = {Bitlis Eren University}, year = {2015}, volume = {5}, pages = {0 - 0}, doi = {10.17678/beujst.41573}, title = {A Partial Solution To An Open Problem}, key = {cite}, author = {KONCA, Şükran} } APA KONCA, Ş . (2015). A Partial Solution To An Open Problem. Bitlis Eren University Journal of Science and Technology, 5 (2), 0-0. Retrieved from http://dergipark.org.tr/beuscitech/issue/40162/477729 MLA KONCA, Ş . "A Partial Solution To An Open Problem". Bitlis Eren University Journal of Science and Technology 5 (2015): 0-0 Chicago KONCA, Ş . "A Partial Solution To An Open Problem". Bitlis Eren University Journal of Science and Technology 5 (2015): 0-0 RIS TY - JOUR T1 - A Partial Solution To An Open Problem AU - Şükran KONCA Y1 - 2015 PY - 2015 N1 - DO - T2 - Bitlis Eren University Journal of Science and Technology JF - Journal JO - JOR SP - 0 EP - 0 VL - 5 IS - 2 SN - -2146-7706 M3 - UR - Y2 - 2019 ER - EndNote %0 Bitlis Eren University Journal of Science and Technology A Partial Solution To An Open Problem %A Şükran KONCA %T A Partial Solution To An Open Problem %D 2015 %J Bitlis Eren University Journal of Science and Technology %P -2146-7706 %V 5 %N 2 %R %U ISNAD KONCA, Şükran . "A Partial Solution To An Open Problem". Bitlis Eren University Journal of Science and Technology 5 / 2 (December 2016): 0-0. AMA KONCA Ş . A Partial Solution To An Open Problem. Bitlis Eren University Journal of Science and Technology. 2015; 5(2): 0-0. Vancouver KONCA Ş . A Partial Solution To An Open Problem. Bitlis Eren University Journal of Science and Technology. 2015; 5(2): 0-0.