Functional Analysis and Applied Optimization in Banach Spaces by Fabio Botelho

Author:Fabio Botelho
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Similar result is valid for the strong topology of the Banach space U so that a functional is strongly lower semicontinuous (l.s.c.) at u ∈ U, if

(10.4)

Corollary 10.1.6.

Every convex l.s.c. function is also w-l.s.c. (weakly lower semicontinuous).

Proof.

The result follows from the fact that the epigraph of F is convex and closed convex sets are weakly closed.

Definition 10.1.7 (Affine Continuous Function).

Let U be a Banach space. A functional is said to be affine continuous if there exist u ∗ ∈ U ∗ and such that

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