Consequences of Completeness

  • Adam BowersEmail author
  • Nigel J. Kalton (deceased)
Part of the Universitext book series (UTX)


The space \(C[0,1]\) of continuous functions on the interval [0,1] can be equipped with many metrics. Two important examples are the metrics arising from the norms

$$\|f\|_\infty = \max_{s \in [0, 1]} |f(s)| \quad\mbox{and}\quad \|f\|_2 = \left(\int_0^1|f(s)|^2\, ds\right)^{1/2},$$

where \(f\in C[0,1]\). The metric arising from the first norm is complete, whereas the metric induced by the second norm is not (i.e., there exist Cauchy sequences that fail to converge). Completeness of a metric is a very profitable property, as we shall see in this chapter. The first theorem we shall meet is a classical result about metric spaces called the Baire Category Theorem. It originated in Baire’s 1899 doctoral thesis, although metric spaces were not formally defined until later.


Baire Category Theorem Profitable Properties Continuous Linear Bijection Uniform Boundedness Principle Open Mapping Theorem 
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Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of Missouri, ColumbiaColumbiaUSA

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