Viewing topic: Ascoli's Theorem

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Ascoli's Theorem

  • Definition: Let X be a space and let (Y, d) be a metric space. Give C(X, Y) the topology of compact convergence; let F be a subset of C(X, Y). a) If F is equicontinuous under d and the set F_d = {f(a)|f in F} has compact closure for each a in X, then F is contained in a compact subspace of C(C, Y). b) The converse holds if X is locally compact Hausdorff.

    Source: Topology (second edition) by James R. Munkres

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