On the Weak* Convergence of Subdifferentials of Convex Functions

Let us assume that a sequence

of proper lower semicontinuous convex functions is bounded on some open subset of a weakly compactly generated Banach space. It is shown that if

\{ f_{n} \}_{n=1}^{\infty } { f n } n = 1 ∞

is a Mosco converging sequence, then for every subgradient

x^* x ∗

of

f f

at

x x

there are subgradients

x^{*}_{n}\in \partial f_{n}(x_{n}) x n ∗ ∈ ∂ f n ( x n )

such that

\{ x^{*}_{n} \}_{n=1}^{\infty } { x n ∗ } n = 1 ∞

is weakly

^* ∗

converging to

x^* x ∗

.