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Functional Spaces 81
5.2 Spaces of measurable functions
In section 2.1, we introduced an inner product and a norm in the space of continuous
functions (see (2.1.8) and (2.1.9)). As we shall see in the sequel, C0 ( Ī) is not a suitable
space for this kind of norm. Besides, in view of the chosen applications, we must to
enrich our space with new functions. Using the approach of section 2.1, this is quite
cumbersome. For instance, including discontinuous functions is not straightforward.
One of the reasons is that property (2.1.3) could not be satisfied in an extended space
(for example, take u vanishing in I, with the exception of one point).
Here, we would like to make this extension in the very large family of measurable
functions. To this purpose, for any measurable function u, we shall denote by Cu the
class of all the functions equivalent to u. Let w : I → R, be a positive continuous
weight function (we prefer to avoid generalizations concerning w). Then, we define the
following functional space:
(5.2.1) L 2 w(I) :=
Cu
u is measurable and
I
u 2
wdx < +∞ .
Although the elements of L 2 w(I) are not functions, we shall not distinguish between u
and Cu, since the integral in (5.2.1) does not recognize the different representatives of a
certain class. So that, due to (5.1.7), we get the implication
(5.2.2) u ∈ L 2
w(I) and u 2 wdx = 0 ⇐⇒ u ≡ 0.
I
The correct interpretation of the right-hand side in (5.2.2) is that a generic representative
of the class of functions associated with u is an element of the class of functions vanishing
almost everywhere. When I is bounded, the inclusion C 0 ( Ī) ⊂ L2 w(I) is similarly
justified.
In L 2 w(I) we can define an inner product and a norm which are the natural extensions
of those given in section 2.1, i.e.
(5.2.3) (u,v) L 2 w (I) :=
I
uv wdx, ∀u,v ∈ L 2 w(I),