On Multi Normed Linear Spaces

Abstract

In the present paper notions of multi linear space and multi normed linear space are presented and some basic properties of such spaces have been investigated.

An Introduction to Multi Inner Product Space

“…Let R 3 be the Euclidean 3-dimensional pace over R. Let X = {5/(a, b, 0) : a, b ∈ R}. Then X is a multi vector space of R 3 over R. Definition 2.19 (Multivectors, [19]). Let X be an Mvector space over a vector space V k .…”

“…Then every multi point of X , i.e., every element of X pt will be called a multivector of X . Definition 2.20 (Multi scalar field, [19]). Let K be a field.…”

“…(N1), (N2), (N3) and (N4) are called norms or axioms. Definition 2.26 (Completeness, [19]). An Mnormed linear space (X , ∥ ∥) is said to be complete if every Cauchy sequence of Mvectors in (X , ∥ ∥) converges to an Mvector of X .…”

“…In our previous papers ( [9], [11], [19]), [10]), we have introduced the notions of multi metric space, multi metric topology, multi linear (vector) space, multi normed linear space and operator on multi normed linear space along with their various properties and several examples and counter examples. In the present paper, we are going to introduce multi complex numbers and multi complex number valued multi inner product on multi linear (vector) space.…”