# Normed and Inner Product Spaces

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A norm on a vector space V over either the field of real numbers or the field of complex numbers is a nonnegative real-valued function | · | : VR such that | v | = 0 iff v = 0, | av | = | a | | v |, and | v + w | ≤ | v | + | w |. A vector space V together with a norm is called a normed vector space or normed linear space. A norm | · | : VR on a vector space V induces a metric d:V×VR on B by d(v,w) = | v - w |.

The norm defined on a normed linear space ( V, | · | ) induces a norm defined on the linear functionals A : VF where F is the field of real numbers or the field of complex numbers by

| A | = sup{ | A x | : | x | = 1 }.

Similarly norms defined on normed linear spaces ( V, | · |V ) and ( W, | · |W ) induce a norm defined on the linear transformations A : VW by

| A | = sup{ | A x |W : | x |V = 1 }.

In particular, given a normed linear space ( V, | · | ), the induced norm defined on the linear algebra of linear transformations A : VV is given by

| A | = sup{ | A x | : | x | = 1 }.

A (Hermitian) inner product defined on a vector space V over the field of complex numbers is a complex-valued function < · , · > : V × VC such that < v , w > = < w , v >* where * denotes the complex conjugate,  < v , v >  ≥  0 and < v , v > = 0 iff v = 0, and iff v = 0, < w, av > = a < w, v >, (this condition is sometimes replaced with the condition < a w, v > = a < w, v >), and < u, v + w > = < u, v > + < u, w > . A vector space V together with a (Hermitian) inner product is called an inner product space or a unitary vector space. The inner product on a vector space V gives rise to a norm defined by | v | = (< v, v > )½.  An inner product on a vector space V over the field of real numbers is simply a positive definite bilinear function on V × V.
 Table of Inner Products and Norms