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Page "Euclidean space" ¶ 37
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metric and topology
In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.
Instead, with the topology of compact convergence, C ( a, b ) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.
Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one.
In topology one considers topologically complete ( or completely metrizable ) spaces, spaces for which there exists at least one complete metric inducing the given topology.
Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology.
If ( V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric ( a notion of distance ) and therefore a topology on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖ u − v ‖.
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology ( although the resulting metric spaces need not be the same ).
Nets are one of the many tools used in topology to generalize certain concepts that may only be general enough in the context of metric spaces.
In topology, an n-sphere is defined as a space homeomorphic to the boundary of an ( n + 1 )- ball ; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric.
The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially.
If the topology of the topological vector space is induced by a metric which is homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.
Second, the real numbers inherit a metric topology from the metric defined above.

metric and on
However, a common and well-accepted metric is the half power points ( i. e. frequency where the power goes down by half its peak value ) on the output vs. frequency curve.
* FIP — Fielding independent pitching: a metric, scaled to resemble an ERA, that focuses on events within the pitcher's control — home runs, walks, and strikeouts
The cosmological principle implies that the metric should be homogeneous and isotropic on large scales, which uniquely singles out the Friedmann – Lemaître – Robertson – Walker metric ( FLRW metric ).
Because the FLRW metric assumes a uniform distribution of mass and energy, it applies to our Universe only on large scales — local concentrations of matter such as our galaxy are gravitationally bound and as such do not experience the large-scale expansion of space.
This project has also been approved by the Executive Board of the United Nations Framework Convention on Climate Change ( UNFCCC ) for reduction of emission of methane gas into the climate and has been registered with a capacity of reducing 108, 686 metric tonnes CO2 equivalent per annum.
Consider on K the metric induced by the uniform distance.
The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.
where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.
In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,
Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.
The current definition, sometimes known as the metric carat, was adopted in 1907 at the Fourth General Conference on Weights and Measures, and soon afterward in many countries around the world.
The centimetre – gram – second system ( abbreviated CGS or cgs ) is a variant of the metric system of physical units based on centimetre as the unit of length, gram as a unit of mass, and second as a unit of time.
The space C 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.
The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.
The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.
But " having distance 0 " is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M with the equivalence class of sequences converging to x ( i. e., the equivalence class containing the sequence with constant value x ).
Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.
Thus differential geometry may study differentiable manifolds equipped with a connection, a metric ( which may be Riemannian, pseudo-Riemannian, or Finsler ), a special sort of distribution ( such as a CR structure ), and so on.
This is a differential manifold with a Finsler metric, i. e. a Banach norm defined on each tangent space.
Finally, one can use the norm to define a metric ( or distance function ) on R < sup > n </ sup > by
Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.
* The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space.

metric and E
The Euclidean structure makes E < sup > n </ sup > an inner product space ( in fact a Hilbert space ), a normed vector space, and a metric space.
More generally, in a metric space ( E, d ), the sphere of center x and radius is the set of points y such that.
Anyone holding a valid Ontario driver's license ( i. e., excluding a motorcycle license ) with a " Z " endorsement can legally drive any air-brake-equipped truck-trailer combination with a registered-or actual-gross-vehicle-weight ( i. e., including towing-and towed-vehicle ) up to 11 metric tonnes, that includes one trailer weighing no more than 4. 6 tonnes if the license falls under the following three classes: Class E ( school bus — maximum 24-passenger capacity or ambulance ), F ( regular bus — maximum 24-passenger capacity or ambulance ) or G ( car, van, or small-truck ).
Exa-( symbol E ) is a prefix in the metric system denoting 10 < sup > 18 </ sup > or.
Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston ; the possibilities, called Thurston model geometries, include the three-sphere S < sup > 3 </ sup >, three-dimensional Euclidean space E < sup > 3 </ sup >, three-dimensional hyperbolic space H < sup > 3 </ sup >, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.
In the Schwarzschild metric, the interval dt < sub > E </ sub > is given by :< ref > See T D Moyer ( 1981a ), " Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference ", Celestial Mechanics 23 ( 1981 ) pages 33-56, equations 2 & 3 at pages 35-6 combined here and divided throughout by c < sup > 2 </ sup >.</ ref >
* E is the energy ( kilowatt-hours per metric or short ton )
The E. U. will review this policy again in 2018 and may then maintain the status quo or set a deadline for mandatory metric only.
A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E. ( In other words, a " size " here is a positive real number, and a subset is of size E if its diameter is less than E .) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > of < i > X < i /> such that S is contained in the union of the n open balls of radius E around the points a < sub > i </ sub >.
Note that the hyperbolic area of E is exactly 1, when computed using the canonical Poincaré metric.
* The price-to-earnings ratio or P / E ratio is the common metric used to assess the relative valuation of equities.
Its 1979 product listed such diverse titles as the metric conversion calculator Metri-Vert, an E. S. P.
A Hermitian metric on a complex vector bundle E over a smooth manifold M is a smoothly varying positive-definite Hermitian form on each fiber.
With respect to some Riemannian metric on M, the restriction of Df to E < sup > s </ sup > must be a contraction and the restriction of Df to E < sup > u </ sup > must be an expansion.
where we are using the standard Cartesian chart for E < sup > 3 </ sup >, with the Euclidean metric tensor

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