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Approximability of operators in constructive metric spaces

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Abstract

The possibility is studied of approximating pointwise defined operators in a broad class of constructive metric spaces. Various ways of representing operators approximately are presented; uniform approximability, approximability, and weak approximability (see Definitions 3.1, 4.2). It is proved that the class of uniformly approximable operators equals the class of uniformly continuous operators. It is also proved that the class of approximable operators equals the class of weakly approximable operators and coincides with the class of operators having the following property: for every natural number n, it is possible to construct a denumerable covering of the domain of the operator by balls such that the operator has oscillation less than 2−n in each ball.

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Abbreviations

NN:

natural number

RN:

rational number

CMS:

constructive metric space

SAS:

simply approximable CMS

uCa:

uniformly C-approximates

uOa:

uniformly O-approximates

Ca:

C-approximates

Oa:

O-approximates

wCa:

weakly C-approximates

wOa:

weakly O-approximates

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Authors
  1. S. V. Pakhomov
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 171–182, 1976. Results announced April 24, 1975.

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Pakhomov, S.V. Approximability of operators in constructive metric spaces. J Math Sci 14, 1539–1546 (1980). https://doi.org/10.1007/BF01693984

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  • DOI: https://doi.org/10.1007/BF01693984

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