Normed Vector Space

A normed vector space is a vector space equipped with a length-measuring function called the norm.

A norm is a function |.| taking as arguments vectors and returning real numbers, satisfying these properties:

1. |a v| = |a| |v| for any scalar a and any vector v
2. |v| > 0 unless v = 0
3. |v + w| ≤ |v| + |w| for all vectors v and w (this is called the triangular inequality, because in a geometrical triangle every side is smaller than the sum of the other two)

Every norm defines a metric in a canonical way: d(x, y) = |x - y|, but not every metric corresponds to a norm. Also, metrics define a topology in the space, and this make the space a topological vector space.