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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.□