Riesz lemma

E746577

mathematical theorem result in functional analysis

Riesz lemma is a fundamental result in functional analysis that characterizes how, in an infinite-dimensional normed space, one can find unit vectors that stay a fixed distance away from any given proper closed subspace.

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All labels observed (1)

Label	Occurrences
Riesz lemma canonical	1

Statements (40)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in functional analysis ⓘ
appliesTo	Banach spaces NERFINISHED ⓘ normed linear spaces ⓘ normed vector spaces ⓘ
assumption	subspace is closed ⓘ subspace is proper ⓘ
conclusion	there exists a unit vector whose distance from the subspace is at least α ⓘ
domainCondition	infinite-dimensional normed space ⓘ proper closed subspace ⓘ
field	functional analysis ⓘ normed vector spaces ⓘ
guaranteesExistenceOf	unit vector at prescribed distance from a proper closed subspace ⓘ
holdsIn	complex normed spaces ⓘ real normed spaces ⓘ
implies	in an infinite-dimensional normed space, the closed unit ball is not compact in the norm topology ⓘ in an infinite-dimensional normed space, the closed unit ball is not sequentially compact ⓘ
involvesConcept	closed subspace ⓘ distance to a subspace ⓘ infinite-dimensionality ⓘ norm ⓘ unit vector ⓘ
mathematicsSubjectClassification	46Axx ⓘ 46Bxx ⓘ
namedAfter	Frigyes Riesz NERFINISHED ⓘ
parameter	real number α with 0 < α < 1 ⓘ
relatedTo	Banach–Alaoglu theorem NERFINISHED ⓘ Hahn–Banach theorem NERFINISHED ⓘ Riesz representation theorem NERFINISHED ⓘ geometric theory of Banach spaces ⓘ
statement	If X is a normed space, Y is a proper closed subspace of X, and 0 < α < 1, then there exists x in X with ∥x∥ = 1 such that the distance from x to Y is at least α. ⓘ
strengthenedBy	various quantitative versions in Banach space theory ⓘ
type	existence lemma NERFINISHED ⓘ
usedFor	constructing basic sequences in Banach spaces ⓘ constructing sequences without convergent subsequences in infinite-dimensional spaces ⓘ demonstrating geometric properties of normed spaces ⓘ proving that closed unit balls in infinite-dimensional normed spaces are not compact ⓘ showing non-compactness of the unit ball in infinite-dimensional Banach spaces ⓘ
usedInProofOf	characterizations of finite-dimensional normed spaces via compactness of the unit ball ⓘ results on basic sequences and Schauder bases ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Frigyes Riesz → knownFor → Riesz lemma ⓘ