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The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C (a, b) of continuous functions on (a, b), for it may contain unbounded functions.
In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε {\displaystyle \varepsilon } - variational principle of Ekeland (1974, 1979).
The distinction between a completely metrizable space and a complete metric space lies in the words there exists at least one metric in the definition of completely metrizable space, which is not the same as there is given a metric (the latter would yield the definition of complete metric space). Once we make the choice of the metric on a ...
Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges. To make this precise: a sequence (x n) in a metric space M is Cauchy if for every ε > 0 there is an integer N such that for all m, n > N, d(x m, x n) < ε.
In a metric space, we can define bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is non-compact but complete; the open interval ...
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
In mathematical analysis, Cauchy completeness can be generalized to a notion of completeness for any metric space. See complete metric space. For an ordered field, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the Archimedean property taken together are equivalent to the others.
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.That is, a topological space (,) is said to be metrizable if there is a metric: [,) such that the topology induced by is . [1] [2] Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.