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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).
Deï¬nition 1.1. A Banach space (over R or C) is a complete normed vector space.If X is a vector space with two diVerent norms, kkand kk 0, then the norms are saidto be equivalent if there are positive constants C 1 and C 2 such that C 1 k x k  k x k 0  C 2 k x kfor all x 2 X.eï¬nition 1.1. A Banach space (over R or C) is a complete normed vector space.If X is a vector space with two diVerent norms, kkand kk 0, then the norms are saidto be equivalent if there are positive constants C 1 and C 2 such that C 1 k x k  k x k 0  C 2 k x kfor all x 2 X.