Abstract
Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C*‐algebras.
Highlights
Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. Such general theorems are inspired by the classical Ascoli theorem, proved by G
As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natural function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct
Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categorical setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory
Summary
Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. We investigate relations between the different sequential function space structures (pointwise convergence, continuous convergence, and uniform convergence) and look at the induced structures on equicontinuous and evenly continuous sets. A subset A of an L-space is called precompact if each sequence in A has a subsequence that converges to a point of X. We investigate limits of sequences of continuous functions in the sequential structures of uniform convergence. A subset H of C(X, Y ) is called equicontinuous at a point x of X if for all U ∈ ᐁ and all sequences ξ X-converging to x,. The following proposition reduces equicontinuity of a set of functions to an ordinary continuity of a suitable function, as one can do in the context of topological spaces [14]. With the equivalences above, this is precisely the condition for H to be equicontinuous in x
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