Attouch-Wets convergence and a differential operator for convex functions

Beer, Gerald; Théra, Michel

doi:10.1090/S0002-9939-1994-1204368-5

Attouch-Wets convergence and a differential operator for convex functions
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by Gerald Beer and Michel Théra PDF

Proc. Amer. Math. Soc. 122 (1994), 851-858 Request permission

Abstract:

The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space X in terms of the behavior of an operator $\Delta$ defined on the space of such functions with values in $X \times R \times {X^ \ast }$, defined by $\Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\}$. We show that $\langle {f_n}\rangle$ is Attouch-Wets convergent to f if and only if points of $\Delta (f)$ lying in a fixed bounded set can be uniformly approximated by points of $\Delta ({f_n})$ for large n. The operator $\Delta$ is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.

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Extensions of Lipschitz functions

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