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From Euclidean to Hilbert Spaces. Edoardo Provenzi
Читать онлайн.Название From Euclidean to Hilbert Spaces
Год выпуска 0
isbn 9781119851301
Автор произведения Edoardo Provenzi
Жанр Математика
Издательство John Wiley & Sons Limited
[1.3]
PROOF.– On one side:
by the triangle inequality, thus ‖v‖ − ‖w‖ ≼ ‖v − w‖. On the other side:
thus ‖w‖ − ‖v‖ ≼ ‖v − w‖, i.e. ‖v‖ − ‖w‖ ≽ − ‖v − w‖.
Hence, −‖v − w‖ ≼ ‖v‖ − ‖w‖ ≼ ‖v − w‖, i.e. |‖v‖ − ‖w‖| ≼ ‖v − w‖.
The following formula is also extremely useful.
THEOREM 1.5 (Carnot’s theorem).– Taking v, w ∈ (V, 〈 , 〉):
and
PROOF.– Direct calculation:
If
The laws presented in this section have immediate consequences which will be highlighted in section 1.2.1.
1.2.1. The parallelogram law and the polarization formula
The parallelogram law in ℝ2 is shown in Figure 1.1. This law can be generalized on a vector space with an arbitrary inner product.
THEOREM 1.6 (Parallelogram law).– Let (V, 〈, 〉) be an inner product space on
Figure 1.1. Parallelogram law in ℝ2: The sum of the squares of the two diagonal lines is equal to two times the sum of the squares of the edges v and w. For a color version of this figure, see www.iste.co.uk/provenzi/spaces.zip
PROOF.– A direct consequence of law [1.4] or law [1.5] taking ‖v + w‖2 then ‖v − w‖2.
□
As we have seen, an inner product induces a norm. The polarization formula can be used to “reverse” roles and write the inner product using the norm.
THEOREM 1.7 (Polarization formula).– Let (V, 〈, 〉) be an inner product space on
and:
PROOF.– This law is a direct consequence of law [1.4], in the real case. For the complex case, w is replaced by iw in law [1.5], and by sesquilinearity, we obtain:
By direct calculation, we can then verify that ‖v + w‖2 − ‖v − w‖2 + i ‖v + iw‖2 − i ‖v − iw‖2 = 4〈v, w〉.
It may seem surprising that something as simple as the parallelogram law may be used to establish a necessary and sufficient condition to guarantee that a norm over a vector space will be induced by an inner product, that is, the norm is Hilbertian. This notion will be formalized in Chapter 4.
1.3. Orthogonal and orthonormal families in inner product spaces
The “geometric” definition of an inner product in ℝ2 and ℝ3 indicates that this product is zero if and only if ϑ, the angle between the vectors, is π/2, which implies cos(ϑ) = 0.
In more complicated vector spaces (e.g. polynomial spaces), or even Euclidean vector spaces of more than three dimensions, it is no longer possible to visualize vectors; their orthogonality must therefore be “axiomatized” via the nullity of their scalar product.
DEFINITION 1.5.– Let (V, 〈, 〉) be a real or complex inner product space of finite dimension n. Let F = {v1, · · · , vn} be a family of vectors in V . Thus:
– F is an orthogonal family of vectors if each different vector pair has an inner product of 0: 〈vi, vj〉 = 0;
– F is an orthonormal family if it is orthogonal and, furthermore, ‖vi‖ = 1 ∀i. Thus, if
An orthonormal family (unit and orthogonal vectors) may be characterized