Search results
Results From The WOW.Com Content Network
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 .
The irrational numbers, with the metric defined by (,) = +, where is the first index for which the continued fraction expansions of and differ (this is a complete metric space) The Cantor set By BCT2 , every finite-dimensional Hausdorff manifold is a Baire space, since it is locally compact and Hausdorff.
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).
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.
An arbitrary product of complete metric spaces is Baire. [23] Every locally compact sober space is a Baire space. [24] Every finite topological space is a Baire space (because a finite space has only finitely many open sets and the intersection of two open dense sets is an open dense set [25]).
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.
A Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function ... The proof is basic in ...
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.