In mathematicsmore specifically in general topologycompactness is a property that generalizes the notion of a subset of Euclidean space being closed i. This notion is defined for more general topological spaces than Euclidean space in various ways. One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.
Thus, if one chooses an infinite number of points in the closed unit interval [0, 1]some of those points will get arbitrarily close to some real number in that space. The same set of points would not accumulate to any point of the open unit interval 0, 1 ; so the open unit interval is not compact.
Euclidean space itself is not compact since it is not bounded. Another example is the definition of distributionswhich uses the space of smooth functions that are zero outside of some unspecified compact space. Various equivalent notions of compactness, including sequential compactness and limit point compactnesscan be developed in general metric spaces. The most useful notion, which is the standard definition of the unqualified term compactnessis phrased in terms of the existence of finite families of open sets that " cover " the space in the sense that each point of the space lies in some set contained in the family.
This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn inexhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally —that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points in the line or plane, for instance has a subsequence that must eventually get arbitrarily close to some other point, called a limit point.
Bolzano's proof relied on the method of bisection : the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts—until it closes down on the desired limit point. The full significance of Bolzano's theoremand its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.
In the s, it became clear that results similar to the Bolzano—Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space.
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuumwhich was seen as fundamental for the rigorous formulation of analysis.
InEduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.
The Heine—Borel theoremas the result is now known, is another special property possessed by closed and bounded sets of real numbers. This property was significant because it allowed for the passage from local information about a set such as the continuity of a function to global information about the set such as the uniform continuity of a function.
This sentiment was expressed by Lebesguewho also exploited it in the development of the integral now bearing his name. Ultimately, the Russian school of point-set topologyunder the direction of Pavel Alexandrov and Pavel Urysohnformulated Heine—Borel compactness in a way that could be applied to the modern notion of a topological space.
It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
Any finite space is trivially compact. A non-trivial example of a compact space is the closed unit interval [0,1] of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open or half-open interval of the real numbers is not compact.
In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary—without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point within the space.
It only takes a minute to sign up. The intuition here is that the positive part of the unit sphere is tiny, so could well be compact even though the whole sphere itself is not. Use Riesz representation theorem. So you're done. Sign up to join this community. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. On the compacity of the space of probability measures Ask Question. Asked 6 years, 9 months ago. Active 6 years, 9 months ago.
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In statisticsmeasures of central tendency and statistical dispersionsuch as the meanmedianand standard deviationare defined in terms of L p metrics, and measures of central tendency can be characterized as solutions to variational problems.
In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the L 1 norm of a solution's vector of parameter values i. Techniques which use an L2 penalty, like ridge regressionencourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L 1 norm and the L 2 norm of the parameter vector. This is a consequence of the Riesz—Thorin interpolation theoremand is made precise with the Hausdorff—Young inequality.
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. In fact, by choosing a Hilbert basis i. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space.
An analogy to this is suggested by taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distancewhich takes into account that streets are either orthogonal or parallel to each other.
The class of p -norms generalizes these two examples and has an abundance of applications in many parts of mathematicsphysicsand computer science. The absolute value bars are unnecessary when p is a rational number and, in reduced form, has an even numerator. The Euclidean norm from above falls into this class and is the 2 -norm, and the 1 -norm is the norm that corresponds to the rectilinear distance.
It turns out that this limit is equivalent to the following definition:. See L -infinity. Abstractly speaking, this means that R n together with the p -norm is a Banach space. This Banach space is the L p -space over R n.
The grid distance or rectilinear distance sometimes called the " Manhattan distance " between two points is never shorter than the length of the line segment between them the Euclidean or "as the crow flies" distance. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:. This fact generalizes to p -norms in that the p -norm x p of any given vector x does not grow with p :.
For the opposite direction, the following relation between the 1 -norm and the 2 -norm is known:. This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy—Schwarz inequality. On the other hand, the formula. It does define an F-normthough, which is homogeneous of degree p. The space of sequences has a complete metric topology provided by the F-norm. Many authors abuse terminology by omitting the quotation marks.In mathematics, a real-valued function is a function whose values are real numbers.
In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable commonly called real functions and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis.
In particular, many function spaces consist of real-valued functions. Measurable functions also form a vector space and an algebra as explained above. Real numbers form a topological space and a complete metric space. Continuous real-valued functions which implies that X is a topological space are important in theories of topological spaces and of metric spaces.
The extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of metric space itself is defined with a real-valued function of two variables, the metricwhich is continuous. The space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a special topological space. Real numbers are used as the codomain to define smooth functions.
A domain of a real smooth function can be the real coordinate space which yields a real multivariable functiona topological vector space an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained aboveand are a subclass of continuous functions.
Though, real-valued L p spaces still have some of the structure explicated above. Each of L p spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes pnamely. Other contexts where real-valued functions and their special properties are used include monotonic functions on ordered setsconvex functions on vector and affine spacesharmonic and subharmonic functions on Riemannian manifoldsanalytic functions usually of one or more real variablesalgebraic functions on real algebraic varietiesand polynomials of one or more real variables.
Weisstein, Eric W.
From Wikipedia, the free encyclopedia. Mathematical function that takes real values. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. See also: Borel function. Main article: Smooth function.
Hidden categories: Articles with short description Short description is different from Wikidata Articles needing additional references from June All articles needing additional references. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.In particular, we shall investigate how the interplay between the under-lying motion the diffusion process corresponding to L and the branch-ing affects the compact support property.
In , the compact support property was shown to be equivalent to a certain analytic criterion con-cerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process.
In a subsequent paper , this analytic property was investigated purely from the point of view of partial differential equations.
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Some of the results obtained in this latter paper yield interesting results concerning the compact sup-port property. In this paper, the results from  that are relevant to the compact support property are presented, sometimes with exten-sions. These results are interwoven with new results and some informal heuristics.
Taken together, they yield a rather comprehensive picture of the compact support property. Inter alia, we show that the concept of a measure-valued process hitting a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion.
It only takes a minute to sign up. Now use the finite intersection property of compact sets. Since you put the homework tag I tried to be a little vague, but let me know if anything is unclear and I'll give detail. Sign up to join this community. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. Intersection of all compact sets of measure 1 in a measure space Ask Question. Asked 9 years ago. Active 9 years ago. Viewed times.
Alex Alex 3, 2 2 gold badges 21 21 silver badges 44 44 bronze badges. Oct 23 '11 at Active Oldest Votes. Nick Strehlke Nick Strehlke 8, 2 2 gold badges 31 31 silver badges 47 47 bronze badges. So, there is no need to use measure theoretic arguments!! Well, I guess in order to conclude you do need to know that the union of finitely many sets of measure zero has measure zero.
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