Graphics Reference
In-Depth Information
=
(
)
1
p
b
p
Ú
||
f
||
f
.
p
a
The L
p
norm satisfies
Fact 2.
(1) || f ||
p
≥ 0.
(2) || f ||
p
= 0 if and only if Ú
a
f = 0.
(3) For every constant c, || cf ||
p
=|c| || f ||
p
.
(4) || f+g ||
p
£ || f ||
p
+ || g ||
p
.
In other words, the norm acts very much like the absolute value function on reals.
We can use it to define a distance function.
If f,g ΠL
p
([a,b]), define the
L
p
Definition.
distance between f and g
, denoted by
d
p
(f,g), by
()
=-
dfg
,
||
f g
||
p
p
The properties listed under Fact 2 above show that d
p
(f,g) is almost a metric on
L
p
([a,b]), but not quite. It is only a pseudometric. It is symmetric and satisfies the tri-
angle inequality, but it is possible to have the distance between two functions be 0
without the functions being equal. If the functions were continuous, then this would
not happen, but we shall see later when we discuss the Fourier transform that the
assumption of continuity would be too restrictive. Therefore, let us use the standard
trick to get a metric from a pseudometric. Consider the equivalence relation ~ on the
space L
p
([a,b]), where
b
Ú
(
)
=
fg f
~
fg
-
0.
a
The function d
p
(f,g) would induce a metric on the set of equivalence classes (see Exer-
cise 5.2.10 in [AgoM05]). With this equivalence relation we are saying that any func-
tion whose integral over [a,b] is zero is treated as if it were identical to the zero
function. In fact, this identification of functions with the corresponding equivalence
class is
always
made. In the future a statement such as “f = g” for two functions f and
g in L
p
([a,b]) technically means that f ~ g. Because one does not want to introduce
new notation, the reader needs to remember that, although we refer to elements of
L
p
([a,b]) as functions, technically
L
p
([a,b])
is really considered to be a set of equivalence classes of functions!
It follows from what has just been said that it is legitimate to call d
p
(f,g) a metric on
L
p
([a,b]). It is called the
L
p
metric
.
(L
p
([a,b]),d
p
) is a complete metric space.
Fact 3.
We concentrate now on the space L
2
([a,b]), also called
Hilbert space
or the space
of
square integrable functions
, which is especially interesting because it admits an
inner product.
