Search results
Results From The WOW.Com Content Network
A net in converges in this topology if and only if it converges pointwise. The topology of pointwise convergence is the same as convergence in the product topology on the space , where is the domain and is the codomain.
If we let be the set of all finite subsets of then the -topology on is called the topology of pointwise convergence. The topology of pointwise convergence on F {\displaystyle F} is identical to the subspace topology that F {\displaystyle F} inherits from Y T {\displaystyle Y^{T}} when Y T {\displaystyle Y^{T}} is endowed with the usual product ...
For example, the strong operator topology on L(X,Y) is the topology of pointwise convergence. For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X: ‖ ‖.
The product topology is also called the topology of pointwise convergence because a sequence (or more generally, a net) in converges if and only if all its projections to the spaces converge.
This (product) topology on is identical to the topology of pointwise convergence. Let E {\displaystyle E} denote the set of all functions f : R → { 0 , 1 } {\displaystyle f:\mathbb {R} \to \{0,1\}} that are equal to 1 {\displaystyle 1} everywhere except for at most finitely many points (that is, such that the set { x : f ( x ) = 0 ...
It is defined as convergence of the sequence of values of the functions at every point. If the functions take their values in a uniform space, then one can define pointwise Cauchy convergence, uniform convergence, and uniform Cauchy convergence of the sequence. Pointwise convergence implies pointwise Cauchy convergence, and the converse holds ...
The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.
Also available is the product topology on the space of set theoretic functions (i.e. not necessarily continuous functions) Y X. In this context, this topology is also referred to as the topology of pointwise convergence. In algebraic topology, the study of homotopy theory is essentially that of discrete invariants of function spaces;