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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).
This is another example of a metric space that is nota normed vector space: V is a metric space, using the metric dened from jjjj, and therefore,according to the above remark, so is C; but Cis not a vector space, so it is not a normed vectorspace. function jjjj: V !R + is a norm on V if it satises(N1) - (N4). A vector space together with a norm is called a normed vector space.Denition: Let Xbe a set. A metric on Xis a function d: X X!R + that satises (D1) -(D4).