In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Browder–Minty theorem (sometimes called the Minty–Browder theorem) states that a
bounded,
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
coercive
Coercion involves compelling a party to act in an involuntary manner through the use of threats, including threats to use force against that party. It involves a set of forceful actions which violate the free will of an individual in order to in ...
and
monotone function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
''T'' from a
real,
separable reflexive Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
''X'' into its
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
''X''
∗ is automatically
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
. That is, for each
continuous linear functional
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
''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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
.)
The theorem is named in honor of
Felix Browder
Felix Earl Browder (; July 31, 1927 – December 10, 2016) was an American mathematician known for his work in nonlinear functional analysis. He received the National Medal of Science in 1999 and was President of the American Mathematical Socie ...
and
George J. Minty, who independently proved it.
See also
*
Pseudo-monotone operator; pseudo-monotone operators obey a near-exact analogue of the Browder–Minty theorem.
References
* (Theorem 10.49)
{{DEFAULTSORT:Browder-Minty theorem
Banach spaces
Theorems in functional analysis
Operator theory