Browder–Minty theorem

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.)

References

Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 364. ISBN 0-387-00444-0. (Theorem 10.49)

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Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Browder–Minty theorem

See also

References