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Discontinuous subnorms

Goldberg, Moshe and Luxemburg, W. A. J. (2001) Discontinuous subnorms. Linear and Multilinear Algebra, 49 (1). pp. 1-24. ISSN 0308-1087. doi:10.1080/03081080108818683.

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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.

Item Type:Article
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Additional Information:© 2001 Taylor & Francis.
Issue or Number:1
Record Number:CaltechAUTHORS:20181003-135909209
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Official Citation:Moshe Goldberg & W. A. J. Luxemburg (2001) Discontinuous subnorms, Linear and Multilinear Algebra, 49:1, 1-24, DOI: 10.1080/03081080108818683
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90102
Deposited By: George Porter
Deposited On:06 Oct 2018 22:17
Last Modified:16 Nov 2021 00:40

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