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Functional Spaces 85
Higher-order derivatives are similarly defined. The measurable function v is the k-th
derivative of u if
(5.4.4)
I
u dk
φ
dx = (−1)k vφ dx, ∀φ ∈ C
dxk I
∞ 0 (I), k ≥ 1.
The weak derivative of a function u is not uniquely determined. However, two different
derivatives are equivalent, hence they are in the same class Cv.
In general, not all integrable functions admit a weak derivative, unless we enlarge
the space of possible derivatives. This extension leads to the definition of the space of
distributions, whose elements are often not even representable as functions (the typical
example is the Dirac delta). For simplicity, we skip this subject and refer the interested
reader to the following authors: schwartz(1966), treves(1967), guelfand, graev
and vilenkin(1970), and garnir, de wilde and schmets(1973).
In the following sections, we study how to use weak derivatives to define additional
functional spaces.
5.5 Transformation of measurable functions in R
Let C denote the set of complex numbers. A function u : R → C is measurable when
both the real part Re(u) and imaginary part Im(u) are measurable. Then we define
(5.5.1) L 2
(R;C) :=
u measurable and
[(Re(u)) 2 + (Im(u)) 2
]dx < +∞ .
Cu
The space in (5.5.1) is a Hilbert space with the (complex) inner product
(5.5.2) (u,v) L2 (R;C) := u¯v dx, u,v ∈ L 2 (R;C),
where ¯v denotes the complex conjugate of v.
R
Let us now consider the operator F : L 2 (R;C) → L 2 (R;C). This maps u ∈ L 2 (R;C)
to the function defined by:
R