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Functional Spaces 83
Every convergent sequence in X is a Cauchy sequence. The converse depends on the
structure of X. This justifies the following definition. A space X, where every Cauchy
sequence converges to an element of X, is called complete space.
Let I be a bounded interval. The space X = C 0 ( Ī) with the norm · w given in
(2.1.9), is not complete. For instance, let Ī = [0,1] and un(x) = xn , x ∈ Ī, n ∈ N.
Therefore, we obtain a Cauchy sequence in C 0 ( Ī) converging in the norm · w, w ≡ 1,
to a discontinuous function.
The correct norm to be used in C0 ( Ī) is defined by (2.5.1). This leads to a complete
space, i.e., Cauchy sequences in C 0 ( Ī) with the norm · ∞ always converge to a
continuous function in that norm (the well-known uniform convergence). Moreover,
L ∞ (I) is also complete with the norm given in (5.2.6). In the following we set (see
(5.2.6))
(5.3.2) u C 0 (Ī) := u L ∞ (I), ∀u ∈ C 0 ( Ī),
provided either I or u are bounded.
Another interesting result is that L 2 w(I) is complete with the norm given in (5.2.4).
When I is bounded, L2 w(I) is the completion of C0 ( Ī) with the norm (2.1.9). This means
that any function in L 2 w(I) can be approximated in the norm of L 2 w(I) by a sequence
of continuous functions. In this case, not only does C 0 ( Ī) ⊂ L2 w(I), but there exists a
constant K > 0 (depending on µ(I) and w) such that
(5.3.3) u L 2 w (I) ≤ K u C 0 (Ī), ∀u ∈ C 0 ( Ī).
A complete space, whose norm is related to an inner product, is called a Hilbert space.
In particular L2 w(I) is a Hilbert space, C0 ( Ī) is not a Hilbert space. More results are
presented in the classical texts of functional analysis.