Гильберт кеңістігінде операторлар жиынындағы топологиялар

Topologies on the set of operators on a Hilbert space
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.

Let be a sequence of linear operators on the Banach space X. Consider the statement that converges to some operator T on X. This could have several different meanings:
If , that is, the operator norm of (the supremum of , where x ranges over the unit ball in X ) converges to 0, we say that in the uniform operator topology. If for all , then we say in the strong operator topology. Finally, suppose that for all x ∈ X we have in the weak topology of X. This means that for all continuous linear functionals F on X. In this case we say that in the weak operator topology.

The most commonly used topologies are the norm, strong, and weak operator topologies. The weak operator topology is useful for compactness arguments, because the unit ball is compact by the Banach–Alaoglu theorem. The norm topology is fundamental because it makes B(H) into a Banach space, but it is too strong for many purposes; for example, B(H) is not separable in this topology. The strong operator topology could be the most commonly used. The ultraweak and ultrastrong topologies are better behaved than the weak and strong operator topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed. For example, the dual space of B(H) in the weak or strong operator topology is too small to have much analytic content. The adjoint map is not continuous in the strong operator and ultrastrong topologies, while the strong* and ultrastrong* topologies are modifications so that the adjoint becomes continuous. They are not used very often. The Arens–Mackey topology and the weak Banach space topology are relatively rarely used. To summarize, the three essential topologies on B(H) are the norm, ultrastrong, and ultraweak topologies. The weak and strong operator topologies are widely used as convenient approximations to the ultraweak and ultrastrong topologies. The other topologies are relatively obscure.