Читать книгу From Euclidean to Hilbert Spaces - Edoardo Provenzi - Страница 15

NOTABLE EXAMPLES.–

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:

thus:

as the two intermediate terms in the penultimate step are zero, since 〈z, w〉 = 〈w, z〉 = 0.

As ‖z‖² ≽ 0, we have seen that:

i.e. |〈v, w〉|² ≼ ‖v‖²‖w‖², hence |〈v, w〉| ≼ ‖v‖‖w‖;

– second proof (in one line!): ∀t ∈ ℝ we have:

The Cauchy-Schwarz inequality allows the concept of the angle between two vectors to be generalized for abstract vector spaces. In fact, it implies the existence of a coefficient k between −1 and +1 such that 〈v, w〉 = ‖v‖‖w‖k, but, given that the restriction of cos to [0, π] creates a bijection with [−1, 1], this means that there is only one ϑ ∈ [0, π] such that 〈v, w〉 = ‖v‖‖w‖ cos ϑ. ϑ ∈ [0, π] is known as the angle between the two vectors v and w.