Databases Reference
In-Depth Information
Example12.3.1:
Another example of a vector space that is of more practical interest to us is the set of all
functions
f
(
t
)
with finite energy. That is,
∞
2
dt
|
f
(
)
|
<
∞
(3)
t
−∞
Let's see if this set constitutes a vector space. If we define additions as pointwise addition and
scalar multiplication in the usual manner, the set of functions
f
(
t
)
obviously satisfies axioms
1, 2, and 4.
If
f
(
t
)
and
g
(
t
)
are functions with finite energy, and
α
is a scalar, then the functions
f
(
t
)
+
g
(
t
)
and
α
f
(
t
)
also have finite energy.
If
f
(
t
)
and
g
(
t
)
are functions with finite energy, then
f
(
t
)
+
g
(
t
)
=
g
(
t
)
+
f
(
t
)
(axiom
1).
If
f
(
t
)
,
g
(
t
)
, and
h
(
t
)
are functions with finite energy, and
α
and
β
are scalars, then
(
f
(
t
)
+
g
(
t
))
+
h
(
t
)
=
f
(
t
)
+
(
g
(
t
)
+
h
(
t
))
and
(αβ)
f
(
t
)
=
α(β
f
(
t
))
(axiom 2).
If
f
(
t
)
,
g
(
t
)
, and
h
(
t
)
are functions with finite energy, and
α
is a scalar, then
α(
f
(
t
)
+
g
(
t
))
=
α
f
(
t
)
+
α
g
(
t
)
and
(α
+
β)
f
(
t
)
=
α
f
(
t
)
+
β
f
(
t
)
(axiom 4).
as the function that is identically zero for
all
t
. This function satisfies the requirement of finite energy, and we can see that axioms 3
and 5 are also satisfied. Finally, if a function
f
Let us define the additive identity function
θ(
t
)
(
t
)
has finite energy, then from Equation (
3
),
the function
also has finite energy, and axiom 6 is satisfied. Therefore, the set of all
functions with finite energy constitutes a vector space. This space is denoted by
L
2
(
−
f
(
t
)
f
)
,or
simply
L
2
.
12.3.3 Subspace
A
subspace S
of a vector space
V
is a subset of
V
whose members satisfy all the axioms of
the vector space. It has the additional property that if
x
and
y
are in
S
, and
α
is a scalar, then
x
+
y
and
α
x
arealsoin
S
.
Example12.3.2:
Consider the set
S
of continuous bounded functions on the interval [0, 1]. Then
S
is a subspace
of the vector space
L
2
.
12.3.4 Basis
One way we can generate a subspace is by taking linear combinations of a set of vectors. If
this set of vectors is
linearly independent
, then the set is called a
basis
for the subspace.
