Inner Product Space - Norms On Inner Product Spaces

Norms On Inner Product Spaces

A linear space with a norm such as:

where p ≠ 2 is a normed space but not an inner product space, because this norm does not satisfy the parallelogram equality required of a norm to have an inner product associated with it.

However, inner product spaces have a naturally defined norm based upon the inner product of the space itself that does satisfy the parallelogram equality:

This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:

• Cauchy–Schwarz inequality: for x, y elements of V

with equality if and only if x and y are linearly dependent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the Cauchy–Bunyakowski–Schwarz inequality.

Because of its importance, its short proof should be noted.

It is trivial to prove the inequality true in the case y = 0. Thus we assume is nonzero, giving us the following:

The complete proof can be obtained by multiplying out this result.

• Orthogonality: The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero vectors x and y in the case = by the identity

We assume the value of the angle is chosen to be in the interval . This is in analogy to the situation in two-dimensional Euclidean space.

In the case =, the angle in the interval is typically defined by

Correspondingly, we will say that non-zero vectors x and y of V are orthogonal if and only if their inner product is zero.

• Homogeneity: for x an element of V and r a scalar

The homogeneity property is completely trivial to prove.

• Triangle inequality: for x, y elements of V

The last two properties show the function defined is indeed a norm.

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.

• Pythagorean theorem: Whenever x, y are in V and ⟨x, y⟩ = 0, then

The proof of the identity requires only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component.

The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated is deeper than the version given above.

An induction on the Pythagorean theorem yields:

• If x₁, ..., x_n are orthogonal vectors, that is, for distinct indices j, k, then

In view of the Cauchy-Schwarz inequality, we also note that is continuous from V × V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:

• Parseval's identity: Suppose V is a complete inner product space. If {x_k} are mutually orthogonal vectors in V then

provided the infinite series on the left is convergent. Completeness of the space is needed to ensure that the sequence of partial sums

which is easily shown to be a Cauchy sequence, is convergent.

• Parallelogram law: for x, y elements of V,

The Parallelogram law is, in fact, a necessary and sufficient condition for the existence of a scalar product corresponding to a given norm. If it holds, the scalar product is defined by the polarization identity:

which is a form of the law of cosines.