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The \(n\)-dual space of the space of \(p\)-summable sequences. (English) Zbl 1289.46039

Summary: In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an \(n\)-normed space, we are interested in bounded multilinear \(n\)-functionals and \(n\)-dual spaces. The concept of bounded multilinear \(n\)-functionals on an \(n\)-normed space was initially introduced by A. G. White jun. [Math. Nachr. 42, 43–60 (1969; Zbl 0185.20003)] and studied further by H. Batkunde, H. Gunawan and Y. E. P. Pangalela [“Bounded linear functionals on the \(n\)-normed space of \(p\)-summable sequences”, Acta Univ. M. Belii, Ser. Math. 21, 66–75 (2013), http://actamath.savbb.sk/pdf/aumb2107.pdf] and S. M. Gozali et al. [Ann. Funct. Anal. 1, No. 1, 72–79 (2010; Zbl 1208.46006)]. In this paper, we revisit the definition of bounded multilinear \(n\)-functionals, introduce the concept of \(n\)-dual spaces, and then determine the \(n\)-dual spaces of \(\ell ^p \) spaces when these spaces are not only equipped with the usual norm, but also with some \(n\)-norms.

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
46C99 Inner product spaces and their generalizations, Hilbert spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
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