Search results
Results From The WOW.Com Content Network
If Y is a vector subspace of the algebraic dual space of X then we will assume that they are associated with the canonical pairing X, Y . In this case, the weak topology on X (resp. the weak topology on Y), denoted by 𝜎(X,Y) (resp. by 𝜎(Y,X)) is the weak topology on X (resp. on Y) with respect to the canonical pairing X, Y .
A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak-* topology. In fact, the weak* topology on the closed unit ball of the dual of a separable space is metrizable, and thus ...
It is stronger than the weak operator topology. The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h 1), h 2)| for h 1, h 2 ∈ H. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)
An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases. Let τ 1 and τ 2 be two topologies on a set X and let B i (x) be a local base for the topology τ i at x ∈ X for i = 1,2.
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
The definition of weak convergence can be extended to Banach spaces. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Banach space B is said to converge weakly to a point x in B if f ( x n ) → f ( x ) {\displaystyle f(x_{n})\to f(x)} for any bounded linear functional f {\displaystyle f} defined on B {\displaystyle B} , that is, for ...
The strong dual of () is a Fréchet space if and only if so in particular, the strong dual of (), which is the space ′ of distributions on U, is not metrizable (note that the weak-* topology on ′ also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives ′ ()).
In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let be a locally compact Hausdorff space.