a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is...

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function...

topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable...

analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation...

These definitions coincide for subsets of a complete metric space, but not in general. A metric space ( M , d ) {\displaystyle (M,d)} is totally bounded...

completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that...

vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable...

completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate...

them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian...

completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness...

{\textstyle \bigcap _{k=0}^{\infty }C_{k}}} and the proof is complete. ∎ In a complete metric space, the following variant of Cantor's intersection theorem...

which the space is a complete metric space. A Hilbert space is a special case of a Banach space. The earliest Hilbert spaces were studied from this point...

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other...

the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm...

pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced...

form a complete metric space under the metric induced by the inner product defined above. A complete metric space is also called a Cauchy space, because...

every nonempty complete metric space with no isolated point is uncountable. (If X {\displaystyle X} is a nonempty countable metric space with no isolated...

metric space is a metric space where any two points are the endpoints of a minimizing geodesic. Hadamard space is a complete simply connected space with...

e. g., that it is a complete uniform space with respect to an aforementioned uniformity, e. g., that it is a complete metric space with respect to an aforementioned...

product space Kolmogorov space Lp-space Lens space Liouville space Locally finite space Loop space Lorentz space Mapping space Measure space Metric space Minkowski...

set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set...

pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence...

provable The completeness of the real numbers, which implies that there are no "gaps" in the real numbers Complete metric space, a metric space in which every...

iterated function system is a finite set of contraction mappings on a complete metric space. Symbolically, { f i : X → X ∣ i = 1 , 2 , … , N } ,   N ∈ N {\displaystyle...

a Cantor space. Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric space. (To see...

also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete metric space such that, given any points x...

theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise...

the property of a space that every Cauchy sequence converges in the space is called completeness, and is detailed below. A metric space (X, d) in which...

{\displaystyle \mathbb {R} .} But it is nonmeagre in itself, since it is a complete metric space. The set ( [ 0 , 1 ] ∩ Q ) ∪ { 2 } {\displaystyle ([0,1]\cap \mathbb...

above implies that every complete metric space is a Baire space. The exterior of a subset S {\displaystyle S} of a topological space X , {\displaystyle X...