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22 Polynomial Approximation of Differential Equations
A norm · : X → R + is a real positive function in X with the properties:
(2.1.3) u ≥ 0, ∀u ∈ X, u = 0 ⇐⇒ u ≡ 0,
(2.1.4) λu = |λ| u, ∀u ∈ X, ∀λ ∈ R,
(2.1.5) u + v ≤ u + v, ∀u,v ∈ X (triangle inequality).
Inner products and norms are, in general, independent concepts. Nevertheless, whenever
an inner product is available in X, then a norm is automatically defined by setting
(2.1.6) u := (u,u), ∀u ∈ X.
Checking that (2.1.6) gives actually a norm is an easy exercise. In particular (2.1.5) is
a byproduct of the well-known Schwarz inequality
(2.1.7) |(u,v)| ≤ u v, ∀u,v ∈ X.
Detailed proofs of (2.1.7) and of many other properties of inner products and norms are
widely available in all the basic texts of linear algebra.
We give an example. Let X = C0 ( Ī) be the linear space of continuous functions
in the interval Ī. Let w : I → R be a continuous integrable function satisfying w > 0.
Then, when I is bounded, an inner product (· , ·)w and its corresponding norm · w
are defined by
(2.1.8) (u,v)w :=
(2.1.9) uw :=
I
I
uv w dx, ∀u,v ∈ C 0 ( Ī),
u 2 1
w dx
2
, ∀u ∈ C 0 ( Ī).