On the Young-Fenchel transform for convex functions

Beer, Gerald

doi:10.1090/S0002-9939-1988-0937844-8

On the Young-Fenchel transform for convex functions
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Proc. Amer. Math. Soc. 104 (1988), 1115-1123 Request permission

Abstract:

Let $\Gamma (X)$ be the proper lower semicontinuous convex functions on a reflexive Banach space $X$. We exhibit a simple Vietoris-type topology on $\Gamma (X)$, compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from $\Gamma (X)$ to $\Gamma ({X^*})$ 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.

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
Gerald Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), no. 2, 239–253. MR 969914, DOI 10.1017/S0004972700027519
G. A. Beer, C. J. Himmelberg, K. Prikry, and F. S. Van Vleck, The locally finite topology on $2^X$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 168–172. MR 897090, DOI 10.1090/S0002-9939-1987-0897090-2
Sebastiano Francaviglia, Alojzy Lechicki, and Sandro Levi, Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), no. 2, 347–370. MR 813603, DOI 10.1016/0022-247X(85)90246-X
John R. Giles, Convex analysis with application in the differentiation of convex functions, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 650456
Richard B. Holmes, A course on optimization and best approximation, Lecture Notes in Mathematics, Vol. 257, Springer-Verlag, Berlin-New York, 1972. MR 0420367
J.-L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. (9) 52 (1973), 421–441 (1974) (French). MR 500129
Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
E. Michael, Selected Selection Theorems, Amer. Math. Monthly 63 (1956), no. 4, 233–238. MR 1529282, DOI 10.2307/2310346
Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, DOI 10.1016/0022-247X(71)90200-9

Convergence au sens de Mosco; théorie et applications à l’approximation des solutions d’inéquations

David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235. MR 209806, DOI 10.1090/S0002-9939-1967-0209806-6
R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8