The Banach-Alaoglu Theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak-star topology. Named for Stefan Banach and Leonidas Alaoglu, it is a foundational result of functional analysis that supplies a compactness principle in infinite-dimensional settings where the ordinary norm topology offers none.
Facts
StatementThe closed unit ball of the dual space of a normed vector space is compact in the weak* topology. 1 Classification
Statement FormCharacterization Theorem 1 Sources
1. Banach-Alaoglu Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentence
the Banach-Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology
History section
The proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
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