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Definition. A metric space is a set X together with a function d (called a metric or distance function) which assigns a real number d (x, y) to every pair x, y X satisfying the properties (or axioms): 1  d (x, y) 0 and d (x, y) = 0 x = y, 2  d (x, y) = d (y, x), 3  d (x, y) + d (y, z) d (x, z).
A vector space with complete metric coming from a norm is a Banach space. Natural Banach spaces of. functions are many of the most natural function spaces. Other natural function spaces, such as C1[a;b] and Co(R), are not Banach, but still have a metric topology.