Browder-Minty theorem
Encyclopedia
In mathematics
, the Browder–Minty theorem states that a bounded
, continuous
, coercive
and monotone function T from a real
, separable reflexive
Banach space
X into its continuous dual space X∗ is automatically surjective. That is, for each continuous linear functional g ∈ X∗, there exists a solution u ∈ X of the equation T(u) = g. (Note that T itself is not required to be a linear map.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Browder–Minty theorem states that a bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
, continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, coercive
Coercive function
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. More precisely, a function f : Rn → Rn is called coercive if...
and monotone function T from a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, separable reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X into its continuous dual space X∗ is automatically surjective. That is, for each continuous linear functional g ∈ X∗, there exists a solution u ∈ X of the equation T(u) = g. (Note that T itself is not required to be a linear map.)
See also
- Pseudo-monotone operatorPseudo-monotone operatorIn mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator...
; pseudo-monotone operators obey a near-exact analogue of the Browder–Minty theorem.