MATH 304
Linear Algebra
Lecture 26:
Norms and inner products.
Norm
The notion of norm generalizes the notion of length
of a vector in Rn .
Definition. Let V be a vector space. 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 ∈ R (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.
The triangle inequality:
|xi + yi | ≤ |xi | + |yi | ≤ maxj |xj | + maxj |yj |
=⇒ maxj |xj + yj | ≤ maxj |xj | + maxj |yj |
• kxk1 = |x1 | + |x2 | + · · · + |xn |.
Positivity and homogeneity are obvious.
The triangle inequality: |xi + yi | ≤ |xi | + |yi |
P
P
P
=⇒
j |yj |
j |xj | +
j |xj + yj | ≤
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
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Examples. V = C [a, b], f : [a, b] → R.
• kf k∞ = max |f (x)|.
a≤x≤b
• kf k1 =
Z
b
|f (x)| dx.
a
• kf kp =
Z
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
The notion of inner product generalizes the notion
of dot product of vectors in Rn .
Definition. Let V be a 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 ).
Counterexamples. V = R2 .
• hx, yi = x1 y1 − x2 y2 .
Let v = (1, 2), then hv, vi = 12 − 22 = −3.
hx, yi is symmetric and bilinear, but not positive.
• hx, yi = 2x1 y1 + x1 x2 + 2x2 y2 + y1 y2 .
v = (1, 1), w = (1, 0) =⇒ hv, wi = 3, h2v, wi = 8.
hx, yi is positive and symmetric, but not bilinear.
• hx, yi = x1 y1 + x1 y2 − x2 y1 + x2 y2 .
v = (1, 1), w = (1, 0) =⇒ hv, wi = 0, hw, vi = 2.
hx, yi is positive and bilinear, but not symmetric.
Problem. Find an inner product on R2 such that
he1 , e1 i = 2, he2 , e2 i = 3, and he1 , e2 i = −1,
where e1 = (1, 0), e2 = (0, 1).
Let x = (x1 , x2 ), y = (y1 , y2 ) ∈ R2 .
Then x = x1 e1 + x2 e2 , y = y1 e1 + y2 e2 .
Using bilinearity, we obtain
hx, yi = hx1 e1 + x2 e2 , y1 e1 + y2 e2 i
= x1 he1 , y1 e1 + y2 e2 i + x2 he2 , y1 e1 + y2 e2 i
= x1 y1 he1 , e1 i + x1 y2 he1 , e2 i + x2 y1 he2 , e1 i + x2 y2 he2 , e2 i
= 2x1 y1 − x1 y2 − x2 y1 + 3x2 y2 .
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 .