The stability problem of the functional equation was conjectured by Ulam and was solved by Hyers in the case of additive mapping. Baker et al. investigated the superstability of the functional equation from a vector space to real numbers. In this paper, we exhibit the superstability of $m$-additive maps on complete non--Archimedean spaces via a fixed point method raised by Diaz and Margolis.

Normed Space [1, 2, §2]. A norm ‖·‖ on a linear space (U ,F) is a mapping ‖·‖ : U → [0,∞) that satisfies, for all u,v ∈ U , α ∈ F , 1. ‖u‖ = 0 ⇐⇒ u = 0. 2. ‖αu‖ = |α| ‖u‖. 3. Triangle inequality: ‖u+ v‖ ≤ ‖u‖+ ‖v‖. A norm defines a metric d(u,v) := ‖u− v‖ on U . A normed (linear) space (U , ‖·‖) is a linear space U with a norm ‖·‖ defined on it. • The norm is a continuous mapping of U into R+. ...

If X,Y are normed spaces, let B(X,Y ) be the set of all bounded linear maps X → Y . If T : X → Y is a linear map, I take it as known that T is bounded if and only if it is continuous if and only if E ⊆ X being bounded implies that T (E) ⊆ Y is bounded. I also take it as known that B(X,Y ) is a normed space with the operator norm, that if Y is a Banach space then B(X,Y ) is a Banach space, that ...

the stability problem of the functional equation was conjectured by ulam and was solved by hyers in the case of additive mapping. baker et al. investigated the superstability of the functional equation from a vector space to real numbers.in this paper, we exhibit the superstability of $m$-additive maps on complete non--archimedean spaces via a fixed point method raised by diaz and margolis.

Let BY denote the unit ball of a normed linear space Y . A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y , there exists a linear projection P : Y → X such that P (BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlar...

A mapping α from a normed space X into itself is called a Banach operator if there is a constant k such that 0≤ k < 1 and ‖α2(x)−α(x)‖ ≤ k‖α(x)−x‖ for all x ∈X. In this note we study some properties of Banach operators. Among other results we show that if α is a linear Banach operator on a normed space X, then N(α−1) = N((α−1)2), N(α−1)∩R(α−1)= (0) and if X is finite dimensional then X =N(α−1)⊕...

Let BY denote the unit ball of a normed linear space Y . A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y , there exists a linear projection P : Y → X such that P (BY ) ⊂ A. The main results of the paper: (1) Each minimal-volume sufficient enlargem...

Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with th...

Let X be a non empty normed structure and let s1 be a sequence of X. The functor ( ∑ κ α=0(s1)(α))κ∈N yielding a sequence of X is defined as follows: (Def. 1) ( ∑ κ α=0(s1)(α))κ∈N(0) = s1(0) and for every natural number n holds ( ∑ κ α=0(s1)(α))κ∈N(n + 1) = ( ∑ κ α=0(s1)(α))κ∈N(n) + s1(n + 1). One can prove the following proposition (1) Let X be an add-associative right zeroed right complementa...