inner product,
n. 1. Given a real or complex vector space V, an inner product on V is a scalar-valued function on the Cartesian product of V with itself such that:
a) <x + y, z> = <x, z> + <y, z>
b) <cx, y> = c <x, y>
c) <x, y> =
d) <x, x> ≥ 0
e) <x, x> = 0 if and only if x = 0.
For example, in a Euclidean space, the "usual inner product" of two vectors x = <xi > and y = <yi> is the sum, Σ xi yi, of products of the corresponding entries of x and y if the space is real, and is Σ xi i if the space is complex. 2. (for tensors) see metric tensor.
a) <x + y, z> = <x, z> + <y, z>
b) <cx, y> = c <x, y>
c) <x, y> =
d) <x, x> ≥ 0
e) <x, x> = 0 if and only if x = 0.
For example, in a Euclidean space, the "usual inner product" of two vectors x = <xi > and y = <yi> is the sum, Σ xi yi, of products of the corresponding entries of x and y if the space is real, and is Σ xi i if the space is complex. 2. (for tensors) see metric tensor.