Glicksberg fixed-point theorem

E23636

fixed-point theorem theorem in functional analysis

The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.

Statements (30)

Predicate	Object
instanceOf	fixed-point theorem → theorem in functional analysis →
appliesTo	compact convex subsets of locally convex topological vector spaces → set-valued maps →
assumptionOnDomain	nonempty compact convex subset →
assumptionOnMap	nonempty convex compact values → upper semicontinuous set-valued map →
assumptionOnSpace	locally convex topological vector space →
concerns	existence of fixed points →
extends	Kakutani fixed-point theorem →
field	functional analysis → topological vector spaces →
generalizes	Kakutani fixed-point theorem to infinite-dimensional settings →
guarantees	existence of a fixed point for the set-valued map →
hasApplication	existence of Nash equilibria in games with infinitely many strategies →
holdsIn	Hausdorff locally convex topological vector spaces →
isPartOf	fixed-point theory in locally convex spaces →
namedAfter	Irving Glicksberg →
relatedTo	Brouwer fixed-point theorem → Kakutani fixed-point theorem → Schauder fixed-point theorem →
requires	closed graph or upper semicontinuity conditions on the correspondence → compactness of the domain set → convexity of the domain set →
topic	multifunctions → topological fixed-point theory →
usedIn	equilibrium existence proofs → game theory → mathematical economics → noncooperative game theory →

Referenced by (1)

Subject (surface form when different)	Predicate
Kakutani fixed-point theorem →	relatedTo