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As we have seen, projection and decomposition laws are much simpler when an orthonormal basis is available.

Theorem 1.13 states that in a finite-dimensional inner product space, an orthonormal basis can always be constructed from a free family of generators.

THEOREM 1.13.– (The iterative Gram-Schmidt process⁶) If (v₁, . . . , v_n), n ≼ ∞ is a basis of (V, 〈, 〉), then an orthonormal basis of (V, 〈, 〉) can be obtained from (v₁, . . . , v_n).

PROOF.– This proof is constructive in that it provides the method used to construct an orthonormal basis from any arbitrary basis.

– Step 1: normalization of v₁:

– Step 2, illustrated in Figure 1.5: v₂ is projected in the direction of u₁, that is, we consider 〈v₂, u₁〉u₁. We know from theorem 1.12 that the vector difference v₂ − 〈v₂, u₁〉u₁ is orthogonal to u₁. The result is then normalized:

Figure 1.5. Illustration of the second step in the Gram-Schmidt orthonormalization process. For a color version of this figure, see www.iste.co.uk/provenzi/spaces.zip

– Step n, by iteration: