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CNc ; ' c /, and the constraints are preserved and the integrals increased. 6). The goal of these transformations is to “improve” the maximizing sequence so as to get a uniform bound on its continuity. A consequence of these considerations is the following existence result. 11. Suppose that X and Y are compact and c is continuous. Y/, and D ' c . X/ X 'c d : Y Proof. 'n ; n / and improve it, by means of c- and cN -transforms, so that we can assume a uniform bound on the continuity of these functions (the same modulus of continuity as c).

By compactness, we can assume xn ! xN . xn / D `. Nx/ ` since ` is the infimum. Nx/ 2 R, and this value is the minimum of f , realized at xN . 2. Memo: Weak compactness in dual spaces Definition. A sequence xn in a Banach space X is said to be weakly converging to x, and we write xn * x, if for every 2 X 0 (where X 0 is the topological dual of X and h ; i stands for the duality product between these spaces) we have h ; xn i ! h ; xi. A sequence 0 n 2 X is said to be weakly-* converging to , and we write we have h n ; xi !

This set K can be taken compact. Actually, there can be at least two statements: either we want f to be merely continuous on K or we want f to coincide on K with a continuous function defined on X. This theorem is usually stated for real-valued functions, but we happen to need it for functions valued in more general spaces. B/ W B A; B openg). Bi /i of open sets such that any other open set B Y may be expressed as a union of Bi ; for instance, separable metric spaces are second countable). Theorem (weak Lusin).