From Euclidean to Hilbert Spaces. Edoardo Provenzi

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= 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, · · · , v_n} 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: 〈v_i, v_j〉 = 0;

      – F is an orthonormal family if it is orthogonal and, furthermore, ‖v_i‖ = 1 ∀i. Thus, if is an orthogonal family, is an orthonormal family.

      An orthonormal family (unit and orthogonal vectors) may be characterized as follows:

      δ_i,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, . . . , v_n ∈ V . Hence, u is also orthogonal to all vectors in V obtained as a linear combination of v₁, . . . , v_n.

      PROOF.– Let , be an arbitrary linear combination of vectors v₁, . . . , v_n. By direct calculation:

      □

THEOREM 1.8 (Generalized Pythagorean theorem).– Let (V, 〈, 〉) be an inner product space on

. Let u, v ∈ V be orthogonal to each other. Hence:

      More generally, if the vectors v₁,. . . , v_n ∈ 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 (recursion hypothesis);

      – now, we write u = v_n and , so u ⊥ z using Lemma 1.1. Hence, using case n = 2: ‖u + z‖² = ‖u‖² + ‖z‖², but:

      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

and let F be an orthonormal family in V . The distance between any two elements of F is constant and equal to .

      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, · · · , v_n}, v_i ≠ 0 ∀i, then F is free.

      PROOF.– We need to prove the linear independence of the elements v_i, that is, . To this end, we calculate the inner product of the linear combination and an arbitrary vector v_j with j ∈ {1, . . . , n}:

      By hypothesis, none of the vectors in F are zero; the hypothesis that

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