Lecture 25
Andrei Antonenko
April 7, 2003
1 Euclidean spaces
Deﬁnition 1.1.
Let
V
be a vector space. Suppose to any 2 vectors
v,u
∈
V
there assigned a
number from
R
which will be denoted by
h
v,u
i
such that the following 3 properties hold:
Bilinearity
• h
au
1
+
bu
2
,v
i
=
a
h
u
1
,v
i
+
b
h
u
2
,v
i
• h
u,av
1
+
bv
2
i
=
a
h
u,v
1
i
+
b
h
u,v
2
i
Reﬂexivity
h
u,v
i
=
h
v,u
i
Positivity
h
u,u
i ≥
0
; moreover if
h
u,u
i
= 0
, then
u
= 0
.
Any function which satisfy properties above is called a
scalar (inner) product
. A vector
space
V
with a scalar product is called a
(real) Euclidean space
.
Now we will give popular examples of the scalar products in diﬀerent spaces.
R
n
The scalar product of 2 vectors
x
and
y
from
R
n
can be deﬁned as following: if
x
= (
x
1
,x
2
,...,x
n
) and
y
= (
y
1
,y
2
,...,y
n
)
then
h
x,y
i
=
x
1
y
2
+
x
2
y
2
+
···
+
x
n
y
n
.
If the vectors are represented by columnvectors, i.e. by
n
×
1matrices, then
h
x,y
i
=
x
>
y.
Example 1.2.
Let
x
= (1
,
2
,
3)
and
y
= (3
,

1
,
4)
. Then
h
x,y
i
= 1
·
3+2
·
(

1)+3
·
4 = 13
.
Actually, there are other ways of deﬁning a scalar product in the vector space
R
n
. For
example, given positive numbers
a
1
,a
2
,...,a
n
we can deﬁne the scalar product to be
h
(
x
1
,x
2
,...,x
n
)
,
(
y
1
,y
2
,...,y
n
)
i
=
a
1
(
x
1
y
1
) +