MATH 423
Linear Algebra II
Lecture 27:
Norms and inner products.
Euclidean structure
In addition to the linear structure (addition and
scaling), space R3 carries the Euclidean structure:
• length of a vector: |x|,
• angle between vectors: θ,
• dot product: x · y = |x| |y| cos θ.
C
y
θ
A
x
B
Euclidean structure
Properties of vector length:
|x| ≥ 0, |x| = 0 only if x = 0
(positivity)
|r x| = |r | |x|
(homogeneity)
|x + y| ≤ |x| + |y|
(triangle inequality)
y
x
x+y
Euclidean structure
Properties of dot product:
x · x ≥ 0, x · x = 0 only if x = 0
(positivity)
x·y =y·x
(symmetry)
(x + y) · z = x · z + y · z
(distributive law)
(r x) · y = r (x · y)
(homogeneity)
Relations between lengths and dot products:
√
• |x| = x · x
• x · y = 12 |x|2 + |y|2 − |x − y|2
C
x−y
y
θ
A
x
B
Law of cosines:
|x − y|2 = |x|2 + |y|2 − 2|x| |y| cos θ
|x − y|2 = |x|2 + |y|2 − 2 x·y
Norm
The notion of norm generalizes the notion of length
of a vector in Rn .
Definition. Let V be a vector space over F, where
F = R or C. A function α : V → R is called a
norm on V if it has the following properties:
(i) α(x) ≥ 0, α(x) = 0 only for x = 0 (positivity)
(ii) α(r x) = |r | α(x) for all r ∈ F (homogeneity)
(iii) α(x + y) ≤ α(x) + α(y) (triangle inequality)
Notation. The norm of a vector x ∈ V is usually
denoted kxk. Different norms on V are
distinguished by subscripts, e.g., kxk1 and kxk2 .
Examples. V = Rn , x = (x1 , x2 , . . . , xn ) ∈ Rn .
• kxk∞ = max(|x1 |, |x2 |, . . . , |xn |).
Positivity and homogeneity are obvious. Let
x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). Then
x + y = (x1 + y1 , . . . , xn + yn ).
|xi + yi | ≤ |xi | + |yi | ≤ maxj |xj | + maxj |yj |
=⇒ maxj |xj + yj | ≤ maxj |xj | + maxj |yj |
=⇒ kx + yk∞ ≤ kxk∞ + kyk∞ .
• kxk1 = |x1 | + |x2 | + · · · + |xn |.
Positivity and homogeneity are obvious.
The triangle inequality: |xi + yi | ≤ |xi | + |yi |
P
P
P
|x
|
+
|x
+
y
|
≤
=⇒
j
j
j
j |yj |
j
j
Examples. V = Rn , x = (x1 , x2 , . . . , xn ) ∈ Rn .
1/p
• kxkp = |x1 |p + |x2 |p + · · · + |xn |p
, p > 0.
Remark. kxk2 = Euclidean length of x.
Theorem kxkp is a norm on Rn for any p ≥ 1.
Positivity and homogeneity are still obvious (and
hold for any p > 0). The triangle inequality for
p ≥ 1 is known as the Minkowski inequality:
1/p
|x1 + y1 |p + |x2 + y2 |p + · · · + |xn + yn |p
≤
1/p
1/p
.
+ |y1 |p + · · · + |yn |p
≤ |x1 |p + · · · + |xn |p
Normed vector space
Definition. A normed vector space is a vector
space endowed with a norm.
The norm defines a distance function on the normed
vector space: dist(x, y) = kx − yk.
Then we say that a sequence x1 , x2 , . . . converges
to a vector x if dist(x, xn ) → 0 as n → ∞.
Also, we say that a vector x is a good
approximation of a vector x0 if dist(x, x0 ) is small.
Unit circle: kxk = 1
kxk = (x12 + x22 )1/2
1/2
kxk = 21 x12 + x22
kxk = |x1 | + |x2 |
kxk = max(|x1 |, |x2 |)
black
green
blue
red
Theorem 1 Let k · k be an arbitrary norm on Rn . Then
there exist positive constants c1 , c2 such that
c1 kxk2 ≤ kxk ≤ c2 kxk2 for any x ∈ Rn .
Idea of the proof: One shows that the function f (x) = kxk
is continuous on Rn (in the usual sense). Since the Euclidean
unit sphere S n−1 (given by kxk2 = 1) is a closed bounded set,
the function f attains its minimum and maximum values on
S n−1 . These are c1 and c2 in the above inequalities.
Theorem 2 Let k · k1 and k · k2 be arbitrary norms on a
finite-dimensional vector space V . Then there exist positive
constants c1 , c2 such that
c1 kxk2 ≤ kxk1 ≤ c2 kxk2 for any x ∈ V .
Corollary A sequence x1 , x2 , . . . of vectors from V
converges to a vector x ∈ V with respect to the distance
induced by the norm k · k1 if and only if it converges to x with
respect to the distance induced by k · k2 .
Examples. V = C [a, b], f : [a, b] → R.
• kf k∞ = max |f (x)|.
a≤x≤b
b
• kf k1 =
Z
• kf kp =
Z
|f (x)| dx.
a
a
b
p
|f (x)| dx
1/p
, p > 0.
Theorem kf kp is a norm on C [a, b] for any p ≥ 1.
Inner product: real vector space
The notion of inner product generalizes the notion
of dot product of vectors in R3 .
Definition. Let V be a real vector space. A
function β : V × V → R, usually denoted
β(x, y) = hx, yi, is called an inner product on V if
it is positive, symmetric, and bilinear. That is, if
(i) hx, xi ≥ 0, hx, xi = 0 only for x = 0 (positivity)
(ii) hx, yi = hy, xi
(symmetry)
(iii) hr x, yi = r hx, yi
(homogeneity)
(iv) hx + y, zi = hx, zi + hy, zi (distributive law)
An inner product space is a vector space endowed
with an inner product.
Examples. V = Rn .
• hx, yi = x · y = x1 y1 + x2 y2 + · · · + xn yn .
• hx, yi = d1 x1 y1 + d2 x2 y2 + · · · + dn xn yn ,
where d1 , d2 , . . . , dn > 0.
• hx, yi = (Dx) · (Dy),
where D is an invertible n×n matrix.
Remarks. (a) Invertibility of D is necessary to show
that hx, xi = 0 =⇒ x = 0.
(b) The second example is a particular case of the
1/2
1/2
1/2
third one when D = diag(d1 , d2 , . . . , dn ).
Example. V = Mm,n (R), space of m×n matrices.
• hA, Bi = trace (AB t ).
If A = (aij ) and B = (bij ), then hA, Bi =
m P
n
P
i=1 j=1
Examples. V = C [a, b].
Z b
f (x)g (x) dx.
• hf , g i =
a
• hf , g i =
Z
b
f (x)g (x)w (x) dx,
a
where w is bounded, piecewise continuous, and
w > 0 everywhere on [a, b].
w is called the weight function.
aij bij .