For a topological space X, let
C1(X) denote the Banach space of all bounded functions f : X →R such that for
every 𝜖 > 0 the set {x ∈ X : |f(x)|≥ 𝜖} is closed and discrete in X, endowed with the
supremum norm. Using spaces of this form we give a direct proof (Corollary 1.5) of a
result of Dashiell and Lindenstrauss on strict convexity of Banach spaces.
Subsequently we obtain two types of characterizations of analytic metric spaces. The
first (Theorem 2.3) is topological and is based on the set theoretic ordering of the
compact subsets of X; this is related to some results of Christensen and
Talagrand. The second (Theorem 3.1) is functional analytic and is based on
the existence of bounded linear operators between the spaces of the form
C1(X).
