What is weakly compact?
Weakly compact set, a set that has some but not all of the properties of compact sets, for example: Sequentially compact space, a set in which every infinite sequence has a convergent subsequence. Limit point compact, a set in which every infinite subset of X has a limit point.
What is relative compactness?
Relative compactness is another property of interest. Definition: A subset S of a topological space X is relative compact when the closure Cl(x) is compact. Note that relative compactness does not carry over to topological subspaces.
Which is the strongest topology?
The discrete topology is the strongest topology on a set, while the trivial topology is the weakest. Finite sets can have many topologies on them.
What is a compact subspace?
A subset K of a topological space X is said to be compact if it is compact as a subspace (in the subspace topology). That is, K is compact if for every arbitrary collection C of open subsets of X such that , there is a finite subset F of C such that . Compactness is a “topological” property.
Is the convex convex closed set a weakly compact set?
However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact. As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded.
Is the closed unit ball compact in the weak topology?
Moreover, the closed unit ball in a normed space X is compact in the weak topology if and only if X is reflexive . In more generality, let F be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let X be a normed topological vector space over F, compatible with the absolute value in F.
Can a closed set be weakly compact in a Hilbert space?
Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinitely dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn’t contain 0).
Which is a compact operator in functional analysis?
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.