n-Normed Spaces with Norms of Its Quotient Spaces

Abstract

In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on nnormed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an n-normed space.Mathematics Subject Classification (2010). Primary 46B20; Secondary 54B15.

Bounded Linear Functional on n-Normed Spaces through Its Quotient Spaces

“…By using the above construction we get n quotient spaces, each of them has its own norm. We call the collection of ( * , ‖⋅‖ * ) for = 1, … , a class-1 collection [10]. Furthermore, for a fixed ∈ *1, … , + we generalize the above construction by examining \* 1 , … , +.…”

“…Note that using the above construction, we get ( ) quotient spaces. We collect these quotient spaces in a set and name it a class-collection [10]. One can see that for an -normed space, we can construct class collections.…”

“…These norms will be a new viewpoint in investigating some features of -normed spaces. The new viewpoint is simpler compared to the results obtained by some previous researchers (for more details see [5] and [10]). With respect to these norms, we will define -bounded linear functionals for each ∈ *1, … , + .…”

“…A function ‖⋅, … ,⋅‖: → ℝ satisfying the following properties 1) ‖ 1 , … , ‖ ≥ 0; ‖ 1 , … , ‖ = 0 if and only if 1 , … , are linearly dependent, 2) ‖ 1 , … , ‖ is invariant under permutation, 3) ‖ 1 , … , ‖ = | | ‖ 1 , … , ‖, for any ∈ ℝ, 4) ‖ 1 + 1 ′ , 2 , … , ‖ ≤ ‖ 1 , 2 , … , ‖ + ‖ 1 ′ , 2 , … , ‖, is called an -norm on , and the pair ( , ‖⋅, … ,⋅‖) is called an -normed spaces. This concept is studied further by many researchers in later years (see for instance [5]- [9]). Using this concept, we will study bounded linear functionals on an -normed space.…”

Norms on Quotient Spaces of The 2-Inner Product Space

“…This is a generalization of the concept of inner product space. Concept of 2-normed space and 2inner product space it has been developed extensively with various results by many researchers, (see for instance [5][6][7][8][9][10][11]).…”