A Partial Solution To An Open Problem

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.

References

1. Gahler S (1965). Lineare 2-normierte räume, Math. Nachr., 28: 1-43.
2. Batkunde H, Gunawan H, Pangalela YEP (2013). Bounded linear functionals on the n-normed space of p-summable sequences, Acta Univ. M. Belii Ser. Math., 2013: 66-75, ISSN 1338-7111.
3. Gunawan H, Mashadi M (2001). On n-Normed Spaces. Int. J. Math Math Sci., 27 (10): 631–639.
4. Gunawan H, Setya-Budhi W, Mashadi, Gemawati S (2005). On Volumes of n-Dimensional Parallelepipeds in Spaces, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 16: 48–54.
5. Gozali SG, Gunawan H, Neswan O (2010). On n-Norms and Bounded n-Linear Functionals in a Hilbert Space. Ann. Funct. Anal., 1: 72–79.
6. Pangalela YEP, Gunawan H (2013). The n-Dual Space of p-Summable Sequences, Math. Bohemica, 138 No.4, 439-448 .
7. Gunawan H (2002). On n-inner products, n-norms, and the Cauchy-Schwarz inequality, Sci. Math. Japan., 55: 53-60.
8. Gozali SG, Gunawan H, Neswan O (2010). On n-norms and bounded n-linear functionals in a Hilbert space. Ann. Funct. Anal. 1: 72-79.

9. Gunawan H (2001). The space of p-summable sequences and its natural n-norm, Bull. Austral. Math. Soc., 64: 137-147.
10. Milicic PM (1993). On the Gramm-Schmidt projection in normed spaces, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 4, 89-96.