alaoglu theorem造句

• Its importance comes from the Banach Alaoglu theorem.
• The Banach Alaoglu theorem depends on Tychonoff's theorem about infinite products of compact spaces.
• Under the weak topology the bounded sets coincide with the relatively compact sets which leads to the important Bourbaki Alaoglu theorem.
• The weak operator topology is useful for compactness arguments, because the unit ball is compact by the Banach-Alaoglu theorem.
• The "'Bourbaki Alaoglu theorem "'is a generalization by Bourbaki to dual topologies on locally convex spaces.
• Consequently, for normed vector space ( and hence Banach spaces ) the Bourbaki Alaoglu theorem is equivalent to the Banach Alaoglu theorem.
• Consequently, for normed vector space ( and hence Banach spaces ) the Bourbaki Alaoglu theorem is equivalent to the Banach Alaoglu theorem.
• Since the Banach Alaoglu theorem is proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, in particular the axiom of choice.
• By the Banach Alaoglu theorem and the reflexivity of " H ", the closed unit ball " B " is weakly compact.
• It should be cautioned that despite appearances, the Banach Alaoglu theorem does " not " imply that the weak-* topology is locally compact.
• Non-separable Banach spaces cannot embed isometrically in the separable space, but for every Banach space, one can find a Banach & ndash; Alaoglu theorem.
• An important fact about the weak * topology is the Banach Alaoglu theorem : if " X " is normed, then the closed unit ball in " X * " is weak *-reflexive.
• The Delta-compactness theorem is similar to the Banach Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem ( in the non-separable case ) its proof does not depend on the Axiom of Choice.
• The Delta-compactness theorem is similar to the Banach Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem ( in the non-separable case ) its proof does not depend on the Axiom of Choice.
• The Axiom of Choice is equivalent to a fundamental result of point-set topology, Tychonoff's theorem, and also to the conjunction of two fundamental results of functional analysis, the Banach Alaoglu theorem and the Krein Milman theorem.
• Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, the Banach Tarski paradox, the Hahn Banach theorem, the Banach Steinhaus theorem, the Banach-Mazur game, the Banach Alaoglu theorem, and the Banach fixed-point theorem.
• These include theorems about compactness of certain spaces such as the Banach Alaoglu theorem on the weak-* compactness of the unit ball of the dual space of a normed vector space, and the Arzel? Ascoli theorem characterizing the sequences of functions in which every subsequence has a cellular automata.
• Due to the constructive nature of its proof ( as opposed to the general case, which is based on the axiom of choice ), the sequential Banach Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or lower semi-continuity property in the weak * topology.
• It can be shown that every character on " A " is automatically continuous, and hence ? " A " is a subset of the space " A " * of continuous linear functionals on " A "; moreover, when equipped with the relative weak-* topology, ? " A " turns out to be locally compact and Hausdorff . ( This follows from the Banach Alaoglu theorem . ) The space ? " A " is compact ( in the topology just defined ) if and only if the algebra " A " has an identity element.