MATH 423
Linear Algebra II
Lecture 28:
Inner product spaces.
Norm
The notion of norm generalizes the notion of length
of a vector in R3 .
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 |).
• kxkp = |x1 |p + |x2 |p + · · · + |xn |p
1/p
Examples. V = C [a, b], f : [a, b] → R.
• kf k∞ = max |f (x)|.
a≤x≤b
• kf kp =
Z
a
b
|f (x)|p dx
1/p
, p ≥ 1.
, 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.
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
aij bij .
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.
Inner product: complex vector space
Definition. Let V be a complex vector space. A
function β : V × V → C, usually denoted
β(x, y) = hx, yi, is called an inner product on V if
it is positive, conjugate symmetric, and depends
linearly on the first argument. That is, if
(i) hx, xi ≥ 0, hx, xi = 0 only for x = 0 (positivity)
(ii) hx, yi = hy, xi
(conjugate symmetry)
(iii) hr x, yi = r hx, yi
(homogeneity)
(iv) hx + y, zi = hx, zi + hy, zi (distributive law)
Dependence on the second argument:
hx, r y + szi = r hx, yi + shx, zi.
Example. V = Cn .
• hx, yi = x1 y1 + x2 y2 + · · · + xn yn .
If z = r + is, then z = r − is, zz = r 2 + s 2 = |z|2 .
Therefore hx, xi = |x1 |2 + |x2 |2 + · · · + |xn |2 ≥ 0.
Also, hx, yi = x1 y1 + · · · + xn yn = x1 y1 + · · · + xn yn
= x1 y1 + · · · + xn yn = hy, xi.
Examples. V = C ([a, b], C).
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 the weight function w is bounded, piecewise
continuous, and w > 0 everywhere on [a, b].
Theorem Suppose hx, yi is an inner product on a
vector space V . Then
|hx, yi|2 ≤ hx, xihy, yi for all x, y ∈ V .
Proof: For any t ∈ C let vt = x + ty. Then
hvt , vt i = hx + ty, x + tyi = hx, x + tyi + thy, x + tyi
= hx, xi + thx, yi + thy, xi + tthy, yi.
hx, yi
Assume that y 6= 0 and let t = −
. Then
hy, yi
|hx, yi|2
.
hvt , vt i = hx, xi + thy, xi = hx, xi −
hy, yi
Since hvt , vt i ≥ 0 the desired inequality follows.
In the case y = 0, we have hx, yi = hy, yi = 0.
Cauchy-Schwarz Inequality:
p
p
|hx, yi| ≤ hx, xi hy, yi.
Corollary 1 |x · y| ≤ kxk kyk for all x, y ∈ Rn .
Equivalently, for all xi , yi ∈ R,
(x1 y1 + · · · + xn yn )2 ≤ (x12 + · · · + xn2 )(y12 + · · · + yn2 ).
Corollary 2 For any f , g ∈ C [a, b],
Z b
2 Z b
Z
2
f (x)g (x) dx ≤
|f (x)| dx ·
a
a
a
b
|g (x)|2 dx.
Norms induced by inner products
Theorem Suppose hx, yi is anpinner product on a
vector space V . Then kxk = hx, xi is a norm.
Proof: Positivity is obvious. Homogeneity:
p
p
p
kr xk = hr x, r xi = r r hx, xi = |r | hx, xi.
Triangle inequality (follows from Cauchy-Schwarz’s):
kx + yk2 = hx + y, x + yi = hx, x + yi + hy, x + yi
= hx, xi + hx, yi + hy, xi + hy, yi
= hx, xi + 2 Rehx, yi + hy, yi
≤ hx, xi + 2|hx, yi| + hy, yi
≤ kxk2 + 2kxk kyk + kyk2 = (kxk + kyk)2 .
Examples. • The length of a vector in Rn ,
p
kxk = x12 + x22 + · · · + xn2 ,
is the norm induced by the dot product
x · y = x1 y1 + x2 y2 + · · · + xn yn .
• The norm kf k2 =
Z
a
b
|f (x)|2 dx
1/2
on the
vector space C [a, b] is induced by the inner product
Z b
f (x)g (x) dx.
hf , g i =
a
Angle
Since |hx, yi| ≤ kxk kyk, we can define the angle
between nonzero vectors in any real vector space
with an inner product (and induced norm):
hx, yi
∠(x, y) = arccos
.
kxk kyk
Then hx, yi = kxk kyk cos ∠(x, y).
In particular, vectors x and y are orthogonal
(denoted x ⊥ y) if hx, yi = 0.
In a complex inner product space the angle between
vectors is not defined. However the notion of
orthogonality still makes sense.
x+y
x
y
Pythagorean Law:
x ⊥ y =⇒ kx + yk2 = kxk2 + kyk2
Proof:
kx + yk2 = hx + y, x + yi
= hx, xi + hx, yi + hy, xi + hy, yi
= hx, xi + hy, yi = kxk2 + kyk2 .
y
x−y
x+y
x
x
y
Parallelogram Identity:
kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2
Proof: kx+yk2 = hx+y, x+yi = hx, xi + hx, yi + hy, xi + hy, yi.
Similarly, kx−yk2 = hx, xi − hx, yi − hy, xi + hy, yi.
Then kx+yk2 + kx−yk2 = 2hx, xi + 2hy, yi = 2kxk2 + 2kyk2 .
Theorem 1 A norm on a vector space is induced by an inner
product if and only if the Parallelogram Identity holds for this
norm.
Theorem 2 (Polarization Identity) Suppose V is an inner
product space with an inner product h·, ·i and the induced
norm k · k.
(i) If V is a real vector space, then for any x, y ∈ V ,
1
hx, yi = kx + yk2 − kx − yk2 .
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(ii) If V is a complex vector space, then for any x, y ∈ V ,
1
hx, yi = kx + yk2 − kx − yk2 + ikx + iyk2 − ikx − iyk2 .
4