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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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“Almost” implies “near”
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by Robert M. Anderson PDF
Trans. Amer. Math. Soc. 296 (1986), 229-237 Request permission


We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 296 (1986), 229-237
  • MSC: Primary 03C99; Secondary 03H05, 26E35, 90A99
  • DOI:
  • MathSciNet review: 837809