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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
δi,j is the Kronecker delta5.
1.4. Generalized Pythagorean theorem
The Pythagorean theorem can be generalized to abstract inner product spaces. The general formulation of this theorem is obtained using a lemma.
LEMMA 1.1.– Let (V, 〈, 〉) be a real or complex inner product space. Let u ∈ V be orthogonal to all vectors v1, . . . , vn ∈ V . Hence, u is also orthogonal to all vectors in V obtained as a linear combination of v1, . . . , vn.
PROOF.– Let
THEOREM 1.8 (Generalized Pythagorean theorem).– Let (V, 〈, 〉) be an inner product space on
More generally, if the vectors v1,. . . , vn ∈ V are orthogonal, then:
PROOF.– The two-vector case can be proven thanks to Carnot’s formula:
Proof for cases with n vectors is obtained by recursion:
– the case where n = 2 is demonstrated above;
– we suppose that
– now, we write u = vn and
so:
and:
giving us the desired thesis.
Note that the Pythagorean theorem thesis is a double implication if and only if V is real, in fact, using law [1.6] we have that ‖u + v‖2 = ‖u‖2 + ‖v‖2 holds true if and only if ℜ(〈u, v〉) = 0, which is equivalent to orthogonality if and only if V is real.
The following result gives information concerning the distance between any two vectors within an orthonormal family.
THEOREM 1.9.– Let (V, 〈, 〉) be an inner product space on
PROOF.– Using the Pythagorean theorem: ‖u + (−v)‖2 = ‖u‖2 + ‖v‖2 = 2, from the fact that u ⊥ v.□
1.5. Orthogonality and linear independence
The orthogonality condition is more restrictive than that of linear independence: all orthogonal families are free.
THEOREM 1.10.– Let F be an orthogonal family in (V, 〈, 〉), F = {v1, · · · , vn}, vi ≠ 0 ∀i, then F is free.
PROOF.– We need to prove the linear independence of the elements vi, that is,
By hypothesis, none of the vectors in F are zero; the hypothesis that
This holds for any j ∈ {1, . . . , n}, so the orthogonal family F is free.□
Using the general theory of vector spaces in finite dimensions, an immediate corollary can be derived from theorem 1.10.
COROLLARY 1.1.– An orthogonal family of n non-null vectors in a space (V, 〈, 〉) of dimension n is a basis of V .
DEFINITION 1.6.– A family of n non-null orthogonal vectors in a vector space (V, 〈, 〉) of dimension n is said to be an orthogonal basis of V . If this family is also orthonormal, it is said to be an orthonormal basis of V .
The extension of the orthogonal basis concept to inner product spaces of infinite dimensions will be discussed in Chapter 5. For the moment, it is important to note that an orthogonal basis is made up of the maximum number of mutually orthogonal vectors in a vector space. Taking n to represent the dimension of the space V and proceeding by reductio ad absurdum, imagine the existence of another vector u* ∈ V , u ≠ 0, orthogonal to all of the vectors in an orthogonal basis
Note that in order to determine the components of a vector in relation to an arbitrary basis, we must solve a linear system of n equations with n unknown variables. In fact, if v ∈ V is any vector and (ui) i = 1, . . . , n is a basis of V , then the components of v in (ui) are the scalars α1, . . . , αn such that:
where ui,j is the j-th component of vector ui.
However, in the presence of an orthogonal