GPT Normed Flashcards
Definition - Normed Vector Space:
Definition: A normed vector space is a vector space V equipped with a norm function ||·||, which assigns a non-negative real number to each vector in V, satisfying the following properties:
||x|| ≥ 0 for all x in V (Non-negativity)
||x|| = 0 if and only if x = 0 (Definiteness)
||αx|| = |α| ||x|| for all x in V and α in the underlying field (Homogeneity)
||x + y|| ≤ ||x|| + ||y|| for all x, y in V (Triangle inequality)
Lemma - Norm Equivalence:
Lemma: Let ||·|| and ||·||’ be two norms on a vector space V. The norms are said to be equivalent if there exist positive constants c1 and c2 such that for all x in V:
c1 ||x||’ ≤ ||x|| ≤ c2 ||x||’
Theorem - Completeness of Normed Vector Space:
Theorem: Every Cauchy sequence in a normed vector space V converges to a limit that is also in V. In other words, V is complete under the norm ||·||.
Lemma - Boundedness of Normed Vector Space:
Lemma: A normed vector space V is bounded if and only if there exists a positive real number M such that ||x|| ≤ M for all x in V.
Theorem - Hahn-Banach Theorem:
Theorem: The Hahn-Banach theorem states that given a normed vector space V, and a linear functional f defined on a subspace U of V, there exists an extension of f to the whole space V without increasing its norm.
Lemma - Open Mapping Theorem:
Lemma: The open mapping theorem states that if a continuous linear operator T between normed vector spaces V and W is surjective, then T is an open mapping.
Theorem - Banach-Steinhaus
Theorem (Uniform Boundedness Principle):
Theorem: The Banach-Steinhaus theorem states that if a family of bounded linear operators {Tα} indexed by a parameter α, acting between Banach spaces, satisfies that for each x in the underlying space, the sequence {Tα(x)} is bounded, then the family {Tα} is uniformly bounded.
Definition - Dual Space:
Definition: Given a normed vector space V, the dual space V* is defined as the set of all continuous linear functionals on V.
