n‐Inner Product Spaces

Abstract

IntrodrictioitIn 1928 K. MENGER made considerations concerning axioms for an abstract n-dmensional metric ([lo]). The essential development in the theory of higherdimensional metrics, however, began in 1962 where S. GXHLER ([4], [5]) introduced practicable definitions of a %metric and a 2-normed space end started with systematic investigations of these spaces, above all concerning their topological properties. Later these spaces have been used by several authors for the'study of geometrical, analytical and functional analytical questions. Of especial importance are the considerations concerning %inner product spaces ([l], [ 2 ] , [3], [9], [ll]).The results of ([4], [5]) on ?-metric and 2-normed spaces were extended by S. Giirr-LER to n-metric and n-normed spaces, where n is an arbitrary iistural number L 1. The very interesting results made it desirable to stud? z-inner product spaces with arbitrary natural n z 1. This paper concerns with fundamental properties of n-inner products, among it gives a formula, which is an extension of the CAUCHY-BUNISEOWSEI Inequality to arbitrary n. There are studied relations between n-inner products and certain partially symmetric covariant tensors of rank 27% as Tell as relations between n-inner products and n-norms, too.2. , ..., .) we also write (-, .) and (a, b I a2, ..., an) is to be anderstood as the ex-+) This article is a part of the dissertation (Technical University Warsaw) which was written under supervision of Siegfried Giihler.

On approximate homomorphisms of ternary semigroups

“…In 1989, Misiak (see [12]) defined n-normed spaces and studied their properties. The concept of an n-normed space is a generalization of the notions of a classical normed space and of a 2-normed space introduced by Gähler.…”

“…Now, we recall some basic definitions and facts concerning n-normed spaces (for more details and some examples, we refer the reader to [6,7,12,19]). …”

Orlicz strongly convergent sequences with some applications to Fourier series and statistical convergence

“…A satisfactory theory of 2-normed spaces was initially developed by Gähler [11] in the mid of 1960's, while that of c 2018 BISKA Bilisim Technology n-normed spaces one can see in Misiak [18]. Since then, this concept has been studied by many authors and obtained various results (see [12], [13]).…”

Some sequence spaces of Invariant means and lacunary defined by a Musielak-Orlicz function over $n$-normed spaces

“…The concept of 2-normed spaces was initially developed by Gähler [4] in the mid of 1960's, while that of n-normed spaces one can see in Misiak [14]. Since then, many others have studied this concept and obtained various results, see Gunawan ([5], [6]) and Gunawan and Mashadi [7].…”