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 On some nonArchimedean normed linear spaces
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On some nonArchimedean normed linear spaces Robert, Joseph Pierre
Abstract
A class of complete nonArchimedean pseudonormed linear spaces for which the field of scalars has a trivial valuation is introduced; we call these spaces "Vspaces." Vspaces differ from the classical normed linear spaces in that the homogeneity of the norm is replaced by the requirement that llαxll = llxll for all x and all scalars α≠0; the usual triangle inequality is modified to [ Equation omitted ] and it is assumed that the norm of an element is either zero or is equal to ρⁿ for a fixed real ρ > 1 and some integer n. The concept of a "distinguished basis" in a Vspace is defined. By use of a modified form of Riesz's Lemma, it is shown that every Vspace admits a distinguished basis. Each element of a Vspace then has a uniquely determined series expansion in terms of the elements of a given distinguished basis. An analogue of the PaleyWiener Theorem is proved for distinguished bases. Properties of distinguished bases are exploited throughout this work. Linear and nonlinear operators on Vspaces are also studied. In the usual way, a norm is defined under which the set of bounded operators is a Vspace and the set of bounded linear operators is a "Valgebra." A characterization of bounded linear operators is given as well as theorems on spectral decompositions. Under certain assumptions on the expansions of x, y, A, the existence of solutions to equations of the form xz = y in Valgebras, and of the form Ax = y in arbitrary Vspaces is proved. Approximations of the solutions are obtained. A representation theorem for continuous linear functionals on a Vspace is given. This representation uses an analogue of the classical inner product. Examples of Vspaces and Valgebras discussed include spaces of functions from a Hausdorff space to a normed linear space, on which the pseudonorm characterizes the asymptotic behaviour of the functions. Some results of the theory of pure asymptotics are extended to arbitrary Vspaces.
Item Metadata
Title 
On some nonArchimedean normed linear spaces

Creator  
Publisher 
University of British Columbia

Date Issued 
1965

Description 
A class of complete nonArchimedean pseudonormed linear spaces for which the field of scalars has a trivial valuation is introduced; we call these spaces "Vspaces."
Vspaces differ from the classical normed linear spaces in that the homogeneity of the norm is replaced by the requirement that llαxll = llxll for all x and all scalars α≠0; the usual triangle inequality is modified to
[ Equation omitted ]
and it is assumed that the norm of an element is either zero or is equal to ρⁿ for a fixed real ρ > 1 and some integer n.
The concept of a "distinguished basis" in a Vspace is defined. By use of a modified form of Riesz's Lemma, it is shown that every Vspace admits a distinguished basis. Each element of a Vspace then has a uniquely determined series expansion in terms of the elements of a given distinguished basis. An analogue of the PaleyWiener Theorem is proved for distinguished bases. Properties of distinguished bases are exploited throughout this work.
Linear and nonlinear operators on Vspaces are also studied. In the usual way, a norm is defined under which the set of bounded operators is a Vspace and the set of bounded linear operators is a "Valgebra." A characterization of bounded linear operators is given as well as theorems on spectral decompositions.
Under certain assumptions on the expansions of x, y, A, the existence of solutions to equations of the form xz = y in Valgebras, and of the form Ax = y in arbitrary Vspaces is proved. Approximations of the solutions are obtained.
A representation theorem for continuous linear functionals on a Vspace is given. This representation uses an analogue of the classical inner product.
Examples of Vspaces and Valgebras discussed include spaces of functions from a Hausdorff space to a normed linear space, on which the pseudonorm characterizes the asymptotic behaviour of the functions. Some results of the theory of pure asymptotics are extended to arbitrary Vspaces.

Subject  
Genre  
Type  
Language 
eng

Date Available 
20110922

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0302280

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.