Edoardo Provenzi - From Euclidean to Hilbert Spaces

In n, the complex Euclidean inner product is defined by:

where v = ( v 1, v 2, . . . , v n), w = ( w 1, w 2, . . . , w n) ∈ nare written with their components in relation to any given, but fixed, basis in n.

The symbol will be used throughout to represent either ℝ or in the context of properties which are valid independently of the reality or complexity of the inner product.

THEOREM 1.1.– Let ( V , 〈 , 〉) be an inner product space. We have:

1) 〈 v , 0 V〉 = 0 ∀ v ∈ V ;

2) if 〈 u, w 〉 = 〈 v, w 〉 ∀ w ∈ V , then u and v must coincide;

3) 〈 v, w 〉 = 0 ∀ v ∈ V w = 0 V, i.e. the null vector is the only vector which is orthogonal to all of the other vectors.

PROOF.–

1) 〈 v , 0 V〉 = 〈 v , 0 V+ 0 V〉 = 〈 v , 0 V〉 + 〈 v , 0 V〉 by linearity, i.e. 〈 v , 0 V〉 − 〈 v , 0 V〉 = 0 = 〈 v , 0 V〉.

2) 〈 u, w 〉 = 〈 v, w 〉 ∀ w ∈ V implies, by linearity, that 〈 u − v, w 〉 = 0 ∀ w ∈ V and thus, notably, considering w = u − v , we obtain 〈 u − v, u − v 〉 = 0, implying, due to the definite positiveness of the inner product, that u − v = 0 V, i.e. u = v .

3) If w = 0 V, then 〈 v, w 〉 = 0 ∀ v ∈ V using property (1). Inversely, by hypothesis, it holds that 〈 v, w 〉 = 0 = 〈 v , 0 V〉 ∀ v ∈ V , but then property (2) implies that w = 0 V.

Finally, let us consider a typical property of the complex inner product, which results directly from a property of complex numbers.

THEOREM 1.2.– Let ( V , 〈 , 〉) be a complex inner product space. Thus:

PROOF.– Consider any complex number z = a + ib , so − iz = b − ia , hence b = ℑ ( z ) = ℜ (− iz ). Taking z = 〈 v, w 〉, we obtain ℑ (〈 v, w 〉) = ℜ (− i 〈 v, w 〉) = ℜ (〈 v, iw 〉) by sesquilinearity.

If ( V , 〈, 〉) is an inner product space over , then a norm on V can be defined as follows:

Note that ‖ v ‖ is well defined since 〈 v, v 〉 ≽ 0 ∀ v ∈ V . Once a norm has been established, it is always possible to define a distance between two vectors v, w in V : d ( v, w ) = ‖ v − w ‖.

The vector v ∈ V such that ‖ v ‖= 1 is known as a unit vector . Every vector v ∈ V can be normalized to produce a unit vector, simply by dividing it by its norm.

Three properties of the norm, which should already be known, are listed below. Taking any v, w ∈ V , and any α ∈ :

1) ‖ v ‖≽ 0, ‖ v ‖= 0 v = 0 V;

2) ‖ αv ‖= | α |‖ v ‖(homogeneity);

3) ‖ v + w ‖≼ ‖ v ‖+ ‖ w ‖(triangle inequality).

DEFINITION 1.4 (normed vector space).– A normed vector space is a pair ( V , ‖ ‖) given by a vector space V and a function, called a norm , , satisfying the three properties listed above.

A norm ‖ ‖ is Hilbertian if there exists an inner product 〈 , 〉 on V such that .

Canonically, an inner product space is therefore a normed vector space. Counterexamples can be used to show that the reverse is not generally true.

Note that, by definition, 〈 v, v 〉 = ‖ v ‖ ‖ v ‖, but, in general, the magnitude of the inner product between two different vectors is dominated by the product of their norms. This is the result of the well-known inequality shown below.

THEOREM 1.3 (Cauchy-Schwarz inequality).– For all v, w ∈ ( V , 〈 , 〉) we have:

PROOF.– Dozens of proofs of the Cauchy-Schwarz inequality have been produced. One of the most elegant proofs is shown below, followed by the simplest one:

– first proof : if w = 0 V, then the inequality is verified trivially with 0 = 0. If w ≠ 0 V, then we can define , i.e. and we note that: