Ultraweak topology

In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B_*(H) of B(H), the trace class operators on H. In other words it is the weakest topology such that all elements of the predual are continuous (when considered as functions on B(H)).

Relation with the weak (operator) topology

The ultraweak topology is similar to the weak operator topology. For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology.

One problem with the weak operator topology is that the dual of B(H) with the weak operator topology is "too small". The ultraweak topology fixes this problem: the dual is the full predual B_*(H) of all trace class operators. In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient.

The ultraweak topology can be obtained from the weak operator topology as follows. If H₁ is a separable infinite dimensional Hilbert space then B(H) can be embedded in B(H⊗H₁) by tensoring with the identity map on H₁. Then the restriction of the weak operator topology on B(H⊗H₁) is the ultraweak topology of B(H).

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 1/25/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Ultraweak topology

Relation with the weak (operator) topology

See also