Γ-convergence in Metric Spaces

Abstract

In this chapter we study some properties of Γ-limits when X is metrizable, or, more generally, when X is completely regular. In particular we shall prove that an equi-coercive sequence of functions (F h ) Γ-converges to a function F if and only in $$ \mathop{{\min }}\limits_{{x \in X}} \left( {F + G} \right)(x) = \mathop{{\lim }}\limits_{{h \to \infty }} \;\mathop{{\inf }}\limits_{{x \in X}} \;\left( {{F_h} + G} \right)(x) $$ for every non-negative continuous function G: X → R (compare with Theorem 7.8).