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.)

See also

Pseudo-monotone operator; pseudo-monotone operators obey a near-exact analogue of the Browder–Minty theorem.

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)

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev topological vector
Properties	barrelled complete dual (algebraic/topological) locally convex reflexive separable

Theorems

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