Inner product space

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:

<math>{\rm For\ any}\ x\in V,\ \langle x,x\rangle\geq 0,\ {\rm and}\ \langle x,x\rangle=0\ {\rm if\ and\ only\ if}\ x=0.</math>

<math>{\rm For\ any}\ a\in F\ {\rm (i.e., }\ a\ {\rm is\ a\ scalar),\ and\ any}\ x,y,z\in V,\ \langle z,ax+y\rangle=a\langle z,x\rangle+\langle z,y\rangle.</math>

<math>{\rm For\ any}\ x,y,z\in V,\ \langle x,y\rangle=\langle y,x\rangle^*,\ {\rm where}\ a\mapsto a^*\ {\rm is\ complex\ conjugation;}</math>

<math>({\rm if}\ F=\mathbb{R}\ {\rm then}\ \langle x,y\rangle=\langle y,x\rangle).</math>

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:

• Cauchy-Schwarz inequality: |<x, y>| ≤ ||x||·||y|| for any x, y in V with equality if and only if x and y are linearly dependent
• Triangle inequality: ||x + y|| ≤ ||x|| + ||y||

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.

• Parallelogram law: ||x + y||² + ||x - y||² = 2||x||² + 2||y||²
• Pythagorean theorem: Whenever x, y are in V and <x, y> = 0, then ||x||² + ||y||² = ||x+y||².

A induction on Pythagoras yields:

• If x₁, ..., x_n are orthogonal vectors, that is, <x_j, x_k> = 0 whenever j ≠ k, then

∑ ||x_k||² = ||∑ x_k||²

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:

• Parseval's Identity: If x_k are mutually orthogonal vectors in V and if ∑ x_k converges, then

∑ ||x_k||² = ||∑ x_k||²

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

cos(φ) = <x, y> / (||x||·||y||)

in analogy to the situation in Euclidean space.

Several types of maps A : V -> W between inner product spaces are of relevance:

• Linear maps, i.e. A(ax + y) = a A(x) + A(y) for all a in F and all x and y in V.
• Continuous linear maps, i.e. A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set { ||Ax|| : x in V with ||x|| ≤ 1 } is bounded.
• Isometries, i.e. A is linear and <Ax, Ay> = <x, y> for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective.
• Isometrical isomorphisms, i.e. A is an isometry which is surjective (and hence bijective).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic.

See also:

• outer product