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82 Polynomial Approximation of Differential Equations
(5.2.4) u L 2 w (I) :=
I
u 2 1
wdx
2
, ∀u ∈ L 2 w(I).
The property (2.1.3) is now exactly expressed by (5.2.2).
Another functional space is related to the norm introduced in section 2.5. We first
give some definitions. We say that a measurable function u is essentially bounded when
we can find a constant γ ≥ 0 such that |u| ≤ γ almost everywhere. An essentially
bounded function u behaves like a bounded function, except on sets of measure zero.
Thus, we set
(5.2.5) L ∞ (I) :=
The corresponding norm is given by
Cu
u is essentially bounded .
(5.2.6) uL∞ (I) := inf γ ≥ 0
|u| ≤ γ a.e. .
When I is bounded, continuous functions in Ī belong to L∞ (I). In this case, the norm
(5.2.6) is equal to the norm · ∞, presented in section 2.5.
Introductory discussions on the spaces analyzed herein, and their generalizations, are
contained e.g. in vulikh(1963), goffman and pedrick(1965), okikiolu (1971), and
wouk(1979). Other references are given in the following sections.
5.3 Completeness
We point out an important property of the functional spaces introduced above. We first
recall that a sequence {un}n∈N in a normed space X is a Cauchy sequence when, for
any ǫ > 0, we can find N ∈ N such that
(5.3.1) un1 − un2 < ǫ, ∀n1,n2 > N.