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The space of real numbers and the space of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space, with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces .
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.
They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in ...
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 ...
A metric space M is compact if every open cover has a finite subcover (the usual topological definition). A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.) A metric space M is compact if it is complete and ...
Any complete inner product space has an orthonormal basis. The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).
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.