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 sesquilinear operator (oneandahalf linear operator). A sesquilinear 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 conjugatelinear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the braket 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:
Because of the triangle inequality and because of axiom 2, we see that · is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. Every inner product V space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by V and is constructed by completing V.
A induction on Pythagoras yields:
In view of the CauchySchwarz 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 CauchySchwarz inequality is that it is possible to define the angle φ between two nonzero 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:
See also:
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