- Definition 1: Vector space
A linear vector space
S
S
is a collection of "vectors" such that (1) if
f
1
∈S⇒α
f
1
∈S
f
1
S
α
f
1
S
for all scalars αα
(where
α∈R
α
or
α∈C
α
) and (2) if
f
1
∈S
f
1
S
,
f
2
∈S
f
2
S
, then
(
f
1
+
f
2
)∈S
f
1
f
2
S
To define an abstract linear vector space, we need
- A set of things called "vectors" (X
X)
- A set of things called "scalars" (A
A)
- A vector addition operator (
)
- A scalar multiplication operator (*
*)
The operators need to have all the properties of given
below. Closure is usually the most important to show.
If the scalars αα are real,
SS is called a real vector
space.
If the scalars αα are complex,
SS is called a complex
vector space.
If the "vectors" in SS are functions
of a continuous variable, we sometimes call
SS a linear function
space
We define a set
V
V
to be a vector space if
-
x+y=y+x
x
y
y
x
for each
x
x and
y
y in
V
V
-
x+(y+z)=(x+y)+z
x
y
z
x
y
z
for each
x
x,
y
y, and
z
z in
V
V
-
There is a unique "zero vector" such that
x+0=x
x
0
x
for each
x
x in
V
V
- For each
x
x in
V
V there is a unique vector
−x
x
such that
x+−x=0
x
x
0
.
-
1x=x
1
x
x
-
(
c
1
c
2
)
x=
c
1
(
c
2
x
)
(
c
1
c
2
)
x
c
1
(
c
2
x
)
for each
x
x in
V
V and
c
1
c
1
and
c
2
c
2
in
ℂ
ℂ.
-
c(x+y)=cx+cy
c
x
y
c
x
c
y
for each
x
x and
y
y in
V
V and
c
c in
ℂ
ℂ.
-
(
c
1
+
c
2
)x=
c
1
x+
c
2
x
c
1
c
2
x
c
1
x
c
2
x
for each
x
x in
V
V and
c
1
c
1
and
c
2
c
2
in
ℂ
ℂ.
-
Rn=real vector space
n
real vector space
-
Cn=complex vector space
n
complex vector space
-
L
1
R=
ft
ft
∫−∞∞|ft|dt<∞
L
1
f
t
t
f
t
f
t
is a vector space
-
L
∞
R=
ft
ft
f
(
t
)
is bounded
L
∞
f
t
f
(
t
)
is bounded
f
t
is a vector space
-
L
2
R=
ft
ft
∫−∞∞|ft|2dt<∞
=finite energy signals
L
2
f
t
t
f
t
2
f
t
finite energy signals
is a vector space
-
L
2
0
T
=finite energy functions on interval [0,T]
L
2
0
T
finite energy functions on interval [0,T]
-
ℓ
1
Z
ℓ
1
,
ℓ
2
Z
ℓ
2
,
ℓ
∞
Z
ℓ
∞
are vector spaces
-
The collection of functions piecewise constant between the
integers is a vector space
-
ℝ
+
2=
x
0
x
1
x
0
x
1
(
x
0
>0)∧(
x
1
>0)
ℝ
+
2
x
0
x
1
x
0
0
x
1
0
x
0
x
1
is not a vector space.
11∈
ℝ
+
2
1
1
ℝ
+
2
, but
∀α,α<0:α11∉
ℝ
+
2
α
α
0
α
1
1
ℝ
+
2
-
D=∀
z
,|z|≤1:z∈C
D
z
z
1
z
is not a vector space.
(
z
1
=1)∈D
z
1
1
D
,
(
z
2
=i)∈D
z
2
D
, but
(
z
1
+
z
2
)∉D
z
1
z
2
D
,
|
z
1
+
z
2
|=2>1
z
1
z
2
2
1
Vector spaces can be collections of functions, collections
of sequences, as well as collections of traditional
vectors (i.e. finite lists of numbers)
