The norm defined on a normed linear space ( V,  ·  ) induces a norm defined on the linear functionals A : V → F 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 : V → W 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 : V → V 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 complexvalued function < · , · > : V × V → C 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.
