Abstract

For convex functions on Banach space endowed with a uniformly Gâteaux differentiable norm two observations are presented: first, the Moreau envelope of a proper lower semicontinuous convex function is Gâteaux differentiable; second, if the Moreau envelopes of a sequence of lower semicontinuous convex functions {fn}n=1∞\{f_n\}_{n=1}^{\infty} create a pointwise convergent sequence, then Gâteaux derivatives of the envelopes (computed at a given point) are weakly∗^* convergent.

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Keywords

  • Subdifferentials
  • convex functions
  • Attouch's theorem
  • UG-smooth norm
  • Gâteaux differentiability of the Moreau envelope
  • weak convergence of subdifferentials

Mathematics Subject Classification

49J52

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Suggested citation

D. Zagrodny. “On the Weak* Convergence of Gâteaux Derivatives of Upper Envelopes.” Journal of Convex Analysis 33 (2026), No. 3&4, 1095–1104. https://doi.org/10.68381/jca33058

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.