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Vectors, Pure and Applied by T. W. Körner

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14

Vector spaces with distances

14.1 Orthogonal polynomials

After looking at one direction that the algebraist might take, let us look at a direction that the analyst might take.

Earlier we introduced an inner product on ℝn defined by

images

and obtained the properties given in Lemma 2.3.7. We now turn the procedure on its head and define an inner product by demanding that it obey the conclusions of Lemma 2.3.7.

Definition 14.1.1 If U is a vector space over, we say that a function M : U2 → ℝ is an inner product if writing

images

the following results hold ...

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