2-normed algebras-I

Abstract

The class of 2-normed real algebras, defined in [33], as shown in [29] is either void or contains only trivial algebras. In this paper, a new definition of real or complex 2-normed algebras and 2-Banach algebras are proposed. Several examples of such algebras are given.

2-normed algebras-II

“…be the algebra after augmentation of unity. Then as we have observed in [5], (E , ., . ) is a 2-normed algebra with respect to a 1 , a 2 with unity and as (E, ., . )…”

“…In Section 4, we derive, from the results obtained in Section 3, a new and interesting spectral radius formula for an 1-Banach algebra. The results in Sections 2 and 4 vindicate our definitions of a 2-normed and 2-Banach spaces given in [5].…”

“…Let (E, ., . ) be a 2-Banach algebra with respect to a 1 , a 2 over K with unity (If E is without unity we augment unity as in [5]) a 1 , a 2 ∈ A, where A is an algebra with unity over K, E is a subalgebra of A and (A, ., . ) is a 2-normed space.…”

“…This paper being the sequel to our earlier paper, for notations and definitions, we refer to the said paper [5].…”

Strongly Lacunary Ward Continuity in 2-Normed Spaces

“…Since dim⁡ X ≥ 2, for each x ∈ X there exists a y ∈ X such that x and y are linearly independent, and hence by ( i ), p y ( x ) = || x , y || ≠ 0. Thus the locally convex topological vector space induced by the set { p z : z ∈ X } of seminorms is Hausdorff so X is a Hausdorff space [18]. In this section, we investigate the concepts of a strongly lacunary quasi-Cauchy sequence and strongly lacunary ward continuity of a function on a 2-normed space.…”