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...
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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...
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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...
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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...
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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...
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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...
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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...
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completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate...
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Riemannian manifold (redirect from Riemannian metric)
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...
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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...
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{\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...
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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...
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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...
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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...
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pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced...
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Square-integrable function (redirect from L2 space)
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...
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every nonempty complete metric space with no isolated point is uncountable. (If X {\displaystyle X} is a nonempty countable metric space with no isolated...
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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...
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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...
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product space Kolmogorov space Lp-space Lens space Liouville space Locally finite space Loop space Lorentz space Mapping space Measure space Metric space Minkowski...
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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...
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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...
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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...
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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...
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a Cantor space. Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric space. (To see...
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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...
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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...
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Cauchy sequence (category Metric geometry)
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...
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Meagre set (redirect from Nonmeager space)
{\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...
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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...
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