In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to talk about angles and lengths of vectors. Inner product spaces are generalizations of Euclidean space (where the dot product takes the place of the inner product) and are studied in functional analysis.
Formally, an inner product space is a real or complex vector space V together with a map f : V x V → F where F is the ground field (either R or C). We write <x, y> instead of f(x, y) and require that the following axioms be satisfied:
A function that follows the second and third axioms is called a sesqui-linear operator (one-and-a-half linear operator). A sesqui-linear operator which is positive (<x, x> ≥ 0) is called a semi inner product. A function satisying all three axioms is an inner product. Note that many authors require an inner product to be linear in the first and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the bra-ket notation popular in quantum mechanics.
For several examples of inner product spaces, see Hilbert space.
Here and in the sequel, we will write ||x|| for √<x, x>. This is well defined by axiom 1 and is thought of as the length of the vector x. Directly from the axioms, we can conclude the following:
In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V x V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:
Another consequence of the Cauchy-Schwarz inequality is that it is possible to define the angle φ between two non-zero vectors x and y (at least in the case F = R) by writing
Several types of maps A : V -> W between inner product spaces are of relevance: