Discontinuous subnorms

Abstract

Let S be a subset of a finite-dimensional algebra over a field F either R or C so that S is closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0 ≠ a ε S, and f(αa) = for all a ε S and α ε F. If in addition S is closed under raising to powers, and f(am )=f(a)m for all a ε S and m = 1,2,3,⋯, then f shall be called a submodulus. Further, if S is closed under multiplication, then a submodulus f shall be called a modulus if f(ab) = f(a)f(b) for all a,b ε S. Our main purpose in this paper is to construct discontinuous subnorms, submoduli and moduli, on the complex numbers, the quaternions, and on suitable sets of matrices. In each of these cases we discuss the asymptotic behavior and stability properties of the obtained objects.

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