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 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Banach-Alaoglu Theorem (Wikipedia)
Proved By
Source Banach-Alaoglu Theorem (Wikipedia)
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.
- In Branch: Functional Analysis, Lead sentence
Proved By: Stefan Banach, Lead paragraph
In functional analysis and related branches of mathematics, the Banach-Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.