On the Young-Fenchel transform for convex functions

Gerald Beer
1988 Proceedings of the American Mathematical Society  
Let T(X) be the proper lower semicontinuous convex functions on a reflexive Banach space X. We exhibit a simple Vietoris-type topology on T(X), compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from T(X) to T(X*) is a homeomorphism. Our entirely geometric proof of the bicontinuity of the transform halves the length of Mosco's proof of sequential bicontinuity, and produces a stronger result for nonseparable spaces.
doi:10.1090/s0002-9939-1988-0937844-8 fatcat:fdfnvfvnj5btjlg4lwprruppnu