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We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular,
we give sufficient conditions for a real-valued function defined on the product of
a reflexive Banach space and a normed space to represent all bounded linear functionals
of the latter. We also give two applications to singular differential equations.
An, LH, Du, PX, Duc, DM, Tuoc, PV: Lagrange multipliers for functions derivable along directions in a linear subspace. Proceedings of the American Mathematical Society. 133(2), 595–604 (2005). Publisher Full Text
Hayden, TL: The extension of bilinear functionals. Pacific Journal of Mathematics. 22, 99–108 (1967)
Hayden, TL: Representation theorems in reflexive Banach spaces. Mathematische Zeitschrift. 104(5), 405–406 (1968). Publisher Full Text
Megginson, RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics,p. xx+596. Springer, New York, NY, USA (1998)