Inner Product

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

More precisely, for a real vector space, an inner product satisfies the following four properties. Let , , and be vectors and be a scalar, then:

1. .

2. .

3. .

4. and equality holds iff .

The fourth condition says that the quadratic form is a positive definite quadratic form. Related thereto, note that some authors define an inner product to be a function satisfying only the first three of the above conditions with the added (weaker) condition of being (weakly) nondegenerate (i.e., if for all , then ). In such literature, functions satisfying all four such conditions are typically referred to as positive-definite inner products (Ratcliffe 2006), though inner products which fail to be positive-definite are sometimes called indefinite to avoid confusion. For an indefinite inner product, taking the square root of a negative squared length gives an imaginary number. For the Lorentzian form with sign convention used below, such vectors are timelike, while spacelike vectors have positive squared length. Negating the bilinear form reverses these signs. The resulting square roots do not define a norm or a metric. The Lorentzian inner product is an example of an indefinite inner product.

A vector space together with an inner product on it is called an inner product space. The positive-definite definition uses scalars in the real numbers or complex numbers. Over other fields, bilinear forms or sesquilinear forms require separate conventions.

Examples of inner product spaces include:

1. The real numbers , where the inner product is given by

(1)

2. The Euclidean space , where the inner product is given by the dot product

(2)

3. The vector space of continuous real functions on a nondegenerate closed interval , , with inner product

(3)

When given a complex vector space, the third property above is usually replaced by

(4)

where refers to complex conjugation. With this property, the inner product is called a Hermitian inner product and a complex vector space with a Hermitian inner product is called a Hermitian inner product space.

Every positive-definite inner product space is a metric space. The metric is given by

(5)

If this process results in a complete metric space, it is called a Hilbert space. What's more, every positive-definite inner product naturally induces a norm of the form

(6)

whereby it follows that every positive-definite inner product space is also naturally a normed space. As noted above, inner products which fail to be positive-definite yield "metrics" - and hence, "norms" - which are actually something different due to the possibility of failing their respective positivity conditions. For example, -dimensional Lorentzian space (i.e., the inner product space consisting of with the Lorentzian inner product) comes equipped with a metric tensor of the form

(7)

and a squared norm of the form

(8)

for all vectors . In particular, one can have negative squared infinitesimal intervals and squared norms, as well as nonzero vectors whose vector norm is always zero. As such, the metric (respectively, the norm) fails to actually be a metric (respectively, a norm), though they usually are still called such when no confusion may arise.