Superstability of the Cauchy equation with squares in finite-dimensional normed algebras

The Finite Dimensional Normed Linear Space Theorem

The claim that follows, which I have called the nite-dimensional normed linear space theorem, essentially says that all such spaces are topologically R with the Euclidean norm. This means that in many cases the intuition we obtain in R,R, and R by imagining intervals, circles, and spheres, respectively, will carry over into not only higher dimension R but also any vector space that has nite dim...

Euclidean Structure in Finite Dimensional Normed Spaces

Approximate solutions of homomorphisms and derivations of the generalized Cauchy-Jensen functional equation in $C^*$-ternary algebras

In this paper, we prove Hyers-Ulam-Rassias stability of $C^*$-ternary algebra homomorphism for the following generalized Cauchy-Jensen equation $$eta mu fleft(frac{x+y}{eta}+zright) = f(mu x) + f(mu y) +eta f(mu z)$$ for all $mu in mathbb{S}:= { lambda in mathbb{C} : |lambda | =1}$ and for any fixed positive integer $eta geq 2$ on $C^*$-ternary algebras by using fixed poind alternat...

Normed algebras with involution

We show that most of the theory of Hermitian Banach algebras can be proved for normed ∗-algebras without the assumption of completeness. The condition r(x) ≤ p(x) for all x (where p(x) = r(x∗x)1/2 is the Pták function), which is essential in the theory of Hermitian Banach algebras, is replaced for normed ∗-algebras by the condition r(x + y) ≤ p(x) + p(y) for all x, y. In case of Banach ∗-algebr...

Quotients of Finite-dimensional Quasi-normed Spaces

We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to `∞ for some k. We give two results of this nature. The first guarantees a proportional dimensional cubic quotient when the envelope is cubic; the second gives an estimate for the size of a cubic quotient in terms of a measure of nonconvexity of the quasi-norm.