1
⟨
u
,
v
⟩
=
⟨
v
,
u
⟩
when the
field in discussion is
C
.
Property:
Many inner products may be defined on a vector space.
Proof
If
⟨
·, ·
⟩
is an inner product, then
⟨
·, ·
⟩
r
defined by
⟨
u
,
v
⟩
r
=
r
⟨
u
,
v
⟩
is also an inner producr for any
r
> 0.
Definition.
A vector space endowed with a particular inner product
is called an
inner product space
.
Example: The dot product is an inner product on
R
n
.
Example:
V
=
C
([
a
,
b
]) = {
f

f
: [
a
,
b
]
→
R
,
f
is continuous} is a
vector space, and the function
⟨
·, ·
⟩
:
V
×
V
→
R
defined by
∀
f
,
g
∈
V
is an inner product on
V
.
Axiom 1:
f
2
is continuous and nonnegative.
f
≠
0
⇒
f
2
(
t
0
) > 0 for some
t
0
∈
[
a
,
b
].
⇒
f
2
(
t
) >
p
> 0
∀
[
t
0
−
r
/2,
t
0
+
r
/2]
⊆
[
a
,
b
].
⇒
Axioms 2  4: You examine them.
∫
=
⟩
⟨
b
a
dt
t
g
t
f
g
f
)
(
)
(
,
.
0
)
(
,
2
>
⋅
≥
=
⟩
⟨
∫
p
r
dt
t
f
f
f
b
a
Example:
⟨
A
,
B
⟩
= trace(
AB
T
) is the
Frobenius inner product
on
R
n
×
n
.
Axiom 1:
⟨
A
,
A
⟩
= trace(
AA
T
) =
Σ
1
≤
i
,
j
≤
n
(
a
ij
)
2
> 0
∀
A
≠
O
.
Axiom 2: trace(
AB
T
) = trace(
AB
T
)
T
= trace(
BA
T
).
Axioms 3  4: You examine them.
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Definitions.
For any vector
v
in an inner product space
V
, the
norm
or length of
v
is denoted and defined as

v
 =
⟨
v
,
v
⟩
1/2
.
The
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 Spring '09
 Fong
 Linear Algebra, inner product, Inner product space, Frobenius

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