Environmental Engineering Reference
In-Depth Information
2
x T x
It is also easily and directly seen that in a Euclidean space
x
=
x
,
x
=
satisfies the four norm axioms.
Table 4.6 shows real inner products and corresponding induced norms in some
inner product spaces. It is relatively easy to prove that these inner products satisfy
axioms IP 1to IP 3, but proving IP 4 may require in some cases - such as random
variable inner product spaces - deeper mathematics ( e.g. , using the Lebesgue inte-
gral).
Tabl e 4. 6 Inner product spaces
2
Space
x
,
y
x
n
x T y
x T x
Euclidean
X
=
x
·
y
=
a x
a x
2 dt
Continuous real functions of real argument
X
=
C
[
a
,
b
]
(
t
)
y
(
t
)
dt
(
t
)
1 y k x k
1 x k
Finite real valued sequences
X
= {
x 1
, ...,
x n
}
x 2
Random variables (*) with finite expected
values and variance
X : Ω
E
{
xy
}
E
{
}
(∗∗)
x 2
Real stochastic processes (**)
X : Ω
E
{
xy
}
E
{
}
(*)Ω is the outcome set in a probability space
[47, 64].
(**) ℑ is a function space (or sequence space in case of discrete time), e.g. , L 2 or l 2 spaces
[64].
{
Ω
,
F
,
P
}
4.3.1.2 Projection on an Element
In an inner product space, let the projection of y on x be defined as
)=
y
,
x
(
|
=
,
,
Pr
y
x
2 x
y
x u
x u
(4.81)
x
where
x
x u
=
(4.82)
x
has a unit norm. Since the norm is scalar, by IP 21
may be incorporated into
the inner product. This is a generalization of the projection of a vector y onto a
vector x in Euclidean space, i.e. ,
/
x
y T x u
y
=
Pr
(
y
|
x
)=(
)
x u
.
(4.83)
It almost immediately follows that in the general case y
y
x ,since
y
y
,
x
=
y
,
x
y
,
x
(4.84)
and
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