Abstract
In this paper, the various concepts of bounded and continuous n-linear operators in normed spaces as well as in n-normed spaces are discussed. A sufficient condition for the space of n-linear operators to be a Banach space is given.Further, we prove the equality of two different formulae of norms of an n-linear operator .Also we introduce an n-norm on the dual space of a real linear space.
Highlights
A sufficient condition for the space of n-linear operators to be a Banach space is given.Further, we prove the equality of two different formulae of norms of an n-linear operator . we introduce an n-norm on the dual space of a real linear space
A real valued function ., ..., . : Xn → R is called an n-norm on X if the following conditions hold: (1) x1, . . . , xn = 0 iff x1, . . . , xn are linearly dependent. (2) x1, . . . , xn remains invariant under permutations of x1, . . . , xn. (3) αx1, x2, . . . , xn = |α| x1, . . . , xn ∀x1, . . . , xn ∈ X and α ∈ R. (4) x0 + x1, x2, . . . , xn ≤ x0, . . . , xn + x1, . . . , xn for all x0, x1, . . . , xn ∈ X
In this paper we introduce the notion of bounded n-linear operators as further extensions of the corresponding notions in [14] and a new n-norm as a further work of the notions in [12, 13]
Summary
Let X be a real linear space of dimension greater than 1 and ., . be a real valued function on X × X satisfying the following conditions:. ) is called a linear 2-normed space.2-norms are non-negative and x, y + αx = x, y for every x, y ∈ X and α ∈ R. Let X be a real vector space with dimX ≥ n, n is a poitive integer and be equipped with an inner product ., . Notions of boundedness in 2-normed space was introduced by White [15]. Gozali et al introduced the notion of bounded n-linear functionals in n-normed spaces in [6]. Zofia Lewandowska introduced notions of 2-linear operators on 2-normed sets in [9]. In this paper we introduce the notion of bounded n-linear operators as further extensions of the corresponding notions in [14] and a new n-norm as a further work of the notions in [12, 13]
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