Question: Given a metric space X, the C...X denote the set of all bounded, continuous functionals on X. Show that ¢ C(X) is a linear subspace

Given a metric space X, the C...X€ denote the set of all bounded, continuous functionals on X. Show that
€¢ C(X) is a linear subspace of B(X)
€¢ C(X) is closed (in B(X))
€¢ C(X) is a Banach space with the sup norm

For certain applications somewhat weaker or stronger forms of continuity are appropriate or necessary. These generalization are dealt with in the next two sections. Then we extend the notion of continuity to correspondences, where we find that some of the standard equivalences (exercise 2.70) diverge.

llfll = suplf(x)|

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