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2
ORTHOGONALITY
Inner products and orthogonal functions are fundamental concepts in approximation
theory. All the families introduced in the first chapter form an orthogonal set of func-
tions with respect to a suitable inner product. This property will reveal many other
interesting features.
2.1 Inner products and norms
In the following, X will denote a real vector space, i.e. a set in which addition and scalar
multiplication are defined with the usual basic properties (for the unfamiliar reader a
wide bibliography is available; we suggest, for instance, hoffman and kunze(1971),
lowenthal(1975)).
An inner product (· , ·) : X × X → R is a bilinear application. This means that,
for any couple of vectors in X, their inner product is a real number and the application
is linear for each one of the two variables. Moreover, we require symmetry, i.e.
(2.1.1) (u,v) = (v,u), ∀u,v ∈ X,
and the application has to be positive-definite, i.e. (here 0 is the zero of X):
(2.1.2) (u,u) ≥ 0, ∀u ∈ X, (u,u) = 0 ⇐⇒ u ≡ 0.