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2 - Preliminaries

Published online by Cambridge University Press:  06 January 2010

N. L. Carothers
Affiliation:
Bowling Green State University, Ohio
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Summary

We begin with a brief summary of important facts from functional analysis – some with proofs, some without. Throughout, X, Y, and so on, are normed linear spaces over ℝ. If there is no danger of confusion, we will use ∥ · ∥ to denote the norm in any given normed space; if two or more spaces enter into the discussion, we will use ∥ · ∥X, and so forth, to further identify the norm in question.

Continuous Linear Operators

Given a linear map T : XY, recall that the following are equivalent:

  1. (i) T is continuous at 0 ∈ X.

  2. (ii) T is continuous.

  3. (iii) T is uniformly continuous.

  4. (iv) T is Lipschitz; that is, there exists a constant C < ∞ such that ∥ T xT yYCxyX for all x, yX.

  5. (v) T is bounded; that is, there exists a constant C < ∞ such that ∥T xYCxX for all xX.

If a linear map T : XY is bounded, then there is, in fact, a smallest constant C satisfying ∥T xYCxX for all xX. Indeed, the constant

called the norm of T, works; that is, it satisfies ∥T xY ≤ ∥T ∥ ∥xX and it's the smallest constant to do so. Further, it's not hard to see that (2.1) actually defines a norm on the space B(X, Y) of all bounded, continuous, linear maps T : X → Y.

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Publisher: Cambridge University Press
Print publication year: 2004

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  • Preliminaries
  • N. L. Carothers, Bowling Green State University, Ohio
  • Book: A Short Course on Banach Space Theory
  • Online publication: 06 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511614057.003
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  • Preliminaries
  • N. L. Carothers, Bowling Green State University, Ohio
  • Book: A Short Course on Banach Space Theory
  • Online publication: 06 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511614057.003
Available formats
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  • Preliminaries
  • N. L. Carothers, Bowling Green State University, Ohio
  • Book: A Short Course on Banach Space Theory
  • Online publication: 06 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511614057.003
Available formats
×