a norm is a mapping from to the nonnegative real numbers such that precisely when , for all scalars and , and the triangle inequality holds for all . there are many possible norms, and on a finite-dimensional vector space all are equivalent in the sense that for any two norms and there are positive constants and such that for all .
a norm on a vector space is any function with the three properties of vector length: namely, if and then
an inner product defines a norm as , but not all norms come from inner products. the unit sphere for every inner product is smooth (has no corners) while for the norm
defined on , the unit sphere is the perimeter of the square . it has corners and so it does not arise from an inner product.
the vector -norm is defined for real by