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About 10,000 papers for “compact sets”

Compact sets

Shelby Kilmer · Elsevier eBooks · 2026

Cited by 0

Compact Sets

Daniel J. Madden, Jason A. Aubrey · 2017

A set that is closed and bounded has many more strong properties than a set that is just closed or just bounded. Proving many of the properties of these closed and bounded sets requires greater logical power than directly applying the two definitions separately. Mathematicians have developed an amazing and powerful trick that provides a completely different path to such proofs. A theorem is proved in the chapter that says that, for any closed and bounded set, a very nonintuitive trick works in a proof. Compact sets are simply sets that are closed and bounded. The proof of this is long and detailed. When one assumes that a set is compact (or closed and bounded after Heine-Borel is proved), it is often a good idea not to write out the definition of compact for that set. Writing out that definition can result into tricking ourselves into thinking that he/she has a cover of the set.

Cited by 0

Weakly compact sets

Robert C. James · Transactions of the American Mathematical Society · 1964

It has been conjectured that a closed convex subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C [5]. This reduces easily to the case in which C is bounded, and will be answered in the affirmative [Theorem 4] after some preliminary results are established. Following suggestions by Namioka and Peck, the result is then generalized, first to weakly closed subsets of Banach spaces and then to weakly closed subsets of complete locally convex linear spaces. The original motivation for this conjecture was the knowledge that it is true when C is the unit sphere of a separable Banach space [2], which was later extended to arbitrary Banach spaces [3, Theorem 5]. Additional support was given the conjecture when V. L. Klee [5] proved a seemingly related theorem-namely, that if C is a bounded closed non-weakly-compact convex subset of a Banach space, then there is a decreasing sequence {K1} of nonempty closed convex subsets of C such that, for each x E C and each m E [0, 1), the set x + m(C x) meets only finitely many of the sets {Ki}. Iff is a continuous linear functional whose sup on C is M and if there is no x in C for which f(x) = M, then a suitable choice for Klee's sequence {Ki} is to let Kn = C r1 {x:f (x) > M l/n} for each n. It is interesting to note that the conjecture can be verified easily for a bounded closed convex set that is symmetric about an interior point x, since if x is translated to 0, then the convex set as a unit sphere induces a norm for which the new Banach space is isomorphic to the original space. More generally, if 0 is an interior point of a bounded convex set C and K is the closed convex span of Cu ( C), then K is symmetric about 0 and the sup on K of a continuous linear functional is the larger of its sups on C and C. Therefore for convex bodies the conjecture can be established by using the known theorem for unit spheres [3, Theorem 5, p. 215]. The following theorem is a generalization of a characterization of weak compactness of the unit sphere that was useful in [2]. In the proof of this theorem and thereafter, we shall use the convention that a sequence of nonoverlapping

Cited by 141Open access

Shaking compact sets

CAMPI S., COLESANTI A., GRONCHI P. · Use Siena air (University of Siena) · 2001

Cited by 0

Compact sets (II)

Ken Binmore · Cambridge University Press eBooks · 1981

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Weakly Compact Sets

Robert C. James · Transactions of the American Mathematical Society · 1964

Cited by 9

Shaking Compact Sets

Stefano Campi, Andrea Colesanti, Paolo Gronchi · Florence Research (University of Florence) · 2001

We prove some results concerning an anti-symmetrization procedure called 'shaking', applied to convex bodies.

Cited by 3

Compact Multipolar Sets

Kohur Gowrisankaran, Ramasamy Jesuraj · Canadian Mathematical Bulletin · 1992

Abstract It is proved that a compact subset of a finite product of Brelot harmonic spaces is multipolar if it is a locally multipolar set.

Cited by 0Open access

Approximation on compact sets

Stephen J. Gardiner · Cambridge University Press eBooks · 1995

If Ω is an open set in C or Rn(n ≥ 2), then we will use Ω* to denote the Alexandroff, or one-point, compactification of Ω, and will use A to denote the ideal point. Thus Ω* = Ω ∪ {A}, and a set A is open in Ω* if either A is an open subset of Ω or A = Ω*\K, where K is a compact subset of Ω. In the special case where Ω is C or Rn we continue to write ∞ for A.

Cited by 0

Compact Sets And Operatores

N. Faried, Z. Safi, J. Sarsour · مجلة جامعة الأقصى سلسلة العلوم الإنسانية · 2001

Cited by 0