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Contents
1. Normed vector spaces
1.1. Examples of normed vector spaces
1.2. Continuous linear operators
1.3. Equivalent norms
1.4. Banach spaces
1.5. Matrix norms
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2
3
6
9
11
15
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1. Normed vector spaces
Recall that a vector space over a field K is a set X together with two
operations,
X × X → X, (x, y) 7→ x + y
addition:
scalar multiplication: K × X → X, (λ, x) 7→ λ x
which are required to satisfy certain axioms. The elements of X are called
points or vectors and the elements of K are called scalars.
Definition 1.1. A normed vector space is a pair (X, k.k) consisting of a
vector space X (over R or C) and a mapping
k.k : X → R, x 7→ kxk
such that
(i) kxk = 0 if and only if x = 0
(ii) kλ xk = |λ| kxk for all scalars λ and all x ∈ X
(iii) kx + yk ≤ kxk + kyk for all x, y ∈ X
(Triangle inequality)
The mapping k.k is called a norm on X and kxk is called the norm of x.
The norm of a vector is non-negative and we can think of kxk as the length
of a vector x. The mapping d : X × X → R given by
d(x, y) = kx − yk
for all x, y ∈ X
defines a metric on X. In this way every normed vector space can be
regarded as a metric space.
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1.1. Examples of normed vector spaces.
Example 1.2. The Euclidean norm on Rn .
kxk2 =
√
x·x=
q
x21 + · · · + x2n
where x = (x1 , . . . , xn ) ∈ Rn .
Example 1.3. Other norms on Rn .
(a) The 1-norm.
kxk1 = |x1 | + · · · + |xn |
(b) The ∞-norm.
kxk∞ = max {|x1 |, . . . , |xn |}
where x = (x1 , . . . , xn ) ∈ Rn .
Example 1.4. The complex numbers. (C, |.|) is a normed vector space
where |.| is the modulus,
|z| =
p
x2 + y 2 ,
for all z = x + i y ∈ C
Example 1.5. Let C[0, 1] be the set of all continuous real-valued functions
f : [0, 1] → R. Addition and scalar multiplication of functions can be
defined pointwise:
If f and g are real-valued functions on [0, 1] then
(f + g)(x) = f (x) + g(x)
for all x ∈ [0, 1]
If λ is a scalar then
(λ f )(x) = λ (f (x))
for all x ∈ [0, 1]
With these operations C[0, 1] is a vector space over R. The following define
two different norms on C[0, 1],
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(a)
kf k1 =
Z
1
|f (x)| dx
0
(b)
kf k∞ = sup |f (x)|
x∈[0,1]
for all f ∈ C[0, 1].
Example 1.6. Let c0 be the set of all sequences (xj )∞
j=1 of real numbers
which converge to 0. Addition and scalar multiplication of sequences can
be defined componentwise:
∞
If x = (xj )∞
j=1 and y = (yj )j=1 then
x + y = (xj + yj )∞
j=1
If λ is a scalar then
λx = (λ xj )∞
j=1
With these operations c0 is a vector space over R. We can define a norm
on c0 by
kxk = sup |xj |
j
for all points x = (xj )∞
j=1 in c0 .
Example 1.7. Let `2 be the set of all sequences (xj )∞
j=1 of real numbers
which are square-summable. i.e.
∞
X
x2j < ∞
j=1
Addition and scalar multiplication of sequences can be defined componentwise as in the previous example. With these operations `2 is a vector space
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over R. We can define a norm on `2 by
kxk2 =
∞
X
j=1
2
for all points x = (xj )∞
j=1 in ` .
x2j
! 12
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1.2. Continuous linear operators. Let X and Y be vector spaces over
the same field K. A mapping T : X → Y is called a linear operator if
T (x + y) = T (x) + T (y)
T (λ x) = λ T (x)
for all x, y ∈ X and for all scalars λ.
Now suppose X and Y are normed vector spaces. Since we can regard
X and Y as metric spaces, it makes sense to consider continuous linear
operators. The set of all continuous linear operators T : X → Y is denoted
L(X, Y ).
We can define operations of addition and scalar multiplication on L(X, Y )
as follows:
if S, T ∈ L(X, Y ) then define S + T ∈ L(X, Y ) by
(S + T )(x) = S(x) + T (x)
for all x ∈ X
if T ∈ L(X, Y ) and λ is a scalar then define λ T ∈ L(X, Y ) by
(λ T )(x) = λ(T (x))
for all x ∈ X
With these operations L(X, Y ) is a vector space (over the same field as
X and Y ). We denote by L(X) the set of all continuous linear operators
T : X → X. (i.e. L(X) = L(X, X)).
Theorem 1.8. Let T : X → Y be a linear operator between normed vector
spaces (X, k.k) and (Y, k.k). Then T is continuous if and only if there exists
a real number M such that
kT (x)k ≤ M kxk
for all x ∈ X.
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Definition 1.9. The operator norm on L(X, Y ) is defined by
kT kop = sup kT (x)k
kxk=1
for all T ∈ L(X, Y ).
Theorem 1.10. If T ∈ L(X, Y ) then
kT kop = inf {M : kT (x)k ≤ M kxk for all x ∈ X}
=
sup kT (x)k
kxk≤1
= sup
x6=0
kT (x)k
kxk
Note that
(i) kT kop = 0 if and only if T = 0
(ii) kλ T kop = |λ| kT kop
(iii) kS + T kop ≤ kSkop + kT kop
Thus (L(X, Y ), k.kop ) is a normed vector space.
Also note that if T ∈ L(X, Y ) and S ∈ L(Y, Z) then the composition
ST : X → Z is a continuous linear operator (ie. ST ∈ L(X, Z)). The
operator norm is submultiplicative in the sense that
kST kop ≤ kSkop kT kop
In particular, (L(X), k.kop ) is an example of a normed algebra.
Example 1.11. The unilateral shift operator on `2 is defined by
T : `2 → `2 ,
(x1 , x2 , x3 , . . .) 7→ (0, x1 , x2 , x3 , . . .)
This is a continuous linear operator with kT kop = 1.
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Example 1.12. Let C 1 [0, 1] denote the set of continuous real-valued functions f : [0, 1] → R which have continuous derivatives
df
dx
: [0, 1] → R. Then
C 1 [0, 1] is a vector space with pointwise operations of addition and scalar
multiplication (the same operations as C[0, 1]). The supremum norm
kf k∞ = sup |f (x)|
x∈[0,1]
defines a norm on C 1 [0, 1] and C[0, 1]. The differentiation operator
D : C 1 [0, 1] → C[0, 1],
is a linear operator which is not continuous.
f 7→
df
dx
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1.3. Equivalent norms.
Definition 1.13. Two norms k.k1 and k.k2 on a vector space X are said to
be equivalent if there exist positive real numbers a, b such that
akxk1 ≤ kxk2 ≤ bkxk1
for all x ∈ X.
Theorem 1.14. Let X be a vector space. The following statements are
equivalent:
(i) k.k1 and k.k2 are equivalent norms on X,
(ii) k.k1 and k.k2 generate the same topology on X (i.e. the same open
sets),
(iii) A sequence (xj ) in X converges to a point x with respect to k.k1 if and
only if (xj ) converges to x with respect to k.k2 .
The above result tells us that if we want to check continuity of a mapping
or convergence of a sequence then we are allowed to swap our norm for an
equivalent one. This can make calculations easier.
Theorem 1.15. Let X be a finite dimensional vector space. Then all norms
on X are equivalent.
Example 1.16. Let C[0, 1] denote the set of continuous real-valued functions f : [0, 1] → R. Then the norms
kf k1 =
Z
1
|f (x)| dx
0
and
kf k∞ = sup |f (x)|
x∈[0,1]
are not equivalent on C[0, 1].
Using Theorem 1.15 we can prove the following:
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Theorem 1.17. Let (X, k.k) and (Y, k.k) be normed vector spaces and let
T : X → Y be a linear operator. If X is finite dimensional then T is
continuous.
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1.4. Banach spaces.
Definition 1.18. A Banach space is a normed vector space (X, k.k) which
is complete (with respect to the metric d(x, y) = kx − yk).
Theorem 1.19. Every finite dimensional normed vector space is a Banach
space.
In light of Theorem 1.19 we now consider some infinite dimensional Banach spaces.
Definition 1.20. Let (X, d) and (Y, d0 ) be metric spaces. Let (fn )∞
n=1 be
a sequence of mappings fn : X → Y . Then (fn )∞
n=1 is said to converge
pointwise to a mapping f : X → Y if for each x0 ∈ X, given any > 0
there exists N ∈ N such that
d0 (fn (x0 ), f (x0 )) < for all n ≥ N
(Note that N depends on the point x0 ).
The sequence (fn )∞
n=1 is said to converge uniformly to f : X → Y if given
any > 0 there exists N ∈ N such that
d0 (fn (x), f (x)) < for all n ≥ N, and for all x ∈ X
(In this case N does not depend on any point).
Example 1.21. For each n ∈ N consider the continuous function
fn : (0, ∞) → R,
x 7→
1
nx
Then the sequence (fn )∞
n=1 converges pointwise but not uniformly to the
zero function
f : (0, ∞) → R,
x 7→ 0
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Lemma 1.22. (Uniform Limit Theorem) Let (X, d) and (Y, d0 ) be metric
spaces. Let (fn )∞
n=1 be a sequence of continuous mappings fn : X → Y . If
(fn )∞
n=1 converges uniformly to f : X → Y then f is continuous.
Example 1.23. For each n ∈ N consider the continuous function
fn : [0, 1] → R,
x 7→ xn
Then the sequence (fn )∞
n=1 converges pointwise but not uniformly to the
function
f : [0, 1] → R,
Notice that f is not continuous.

 0 0≤x<1
x 7→
 1 x=1
Theorem 1.24. C[0, 1] with the supremum norm
kf k∞ = sup |f (x)|
x∈[0,1]
is a Banach space.
Theorem 1.25. Let (X, k.k) and (Y, k.k) be normed vector spaces. If Y is
a Banach space then L(X, Y ) with the operator norm
kT kop = sup kT (x)k
kxk≤1
is a Banach space.
The sequence space (`2 , k.k2 ) is another example of an infinite dimensional Banach space.
Definition 1.26. Let (X, k.k) be a normed vector space. If (xk )∞
k=1 is a
sequence in X then we say the formal series
∞
X
k=1
xk
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is convergent if the sequence (sm )∞
m=1 of partial sums
m
X
sm =
xk
k=1
converges to a point x ∈ X. In this case we write
x=
∞
X
xk
k=1
If the series
∞
X
kxk k
k=1
converges in R then we say the series
∞
X
xk
k=1
is absolutely convergent.
Theorem 1.27. Let (X, k.k) be a Banach space. Then every absolutely
convergent series in X is convergent in X. Also,
k
∞
X
xk k ≤
k=1
∞
X
kxk k
k=1
Example 1.28. Let (X, k.k) be a Banach space and let T ∈ L(X). Then
the series
∞
X
1 k
T
e =
k!
k=0
T
is convergent in L(X). In particular, the exponential of a square matrix can
be used to solve systems of differential equations.
Let (X k.k) be a normed vector space. The identity operator I : X → X,
x 7→ x is a multiplicative identity element in L(X) since
T I = I T = I,
for all T ∈ L(X)
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An element T ∈ L(X) is called invertible if there exists S ∈ L(X) such that
T S =ST =I
We call S the inverse of T and we write T −1 = S.
Theorem 1.29. Let (X, k.k) be a Banach space.
(i) If T ∈ L(X) and kT k < 1 then (I − T ) is invertible in L(X) and
(I − T )−1 =
∞
X
Tk
k=0
(ii) The set Inv(L(X)) of all invertible elements in L(X) is an open set
in (L(X), k.kop ).
(iii) The inverse mapping Inv(L(X)) → Inv(L(X)), T 7→ T −1 is continuous.
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1.5. Matrix norms. The collection M m×n (R) of all (m × n)-matrices A =
(aij ) with entries aij in R is a finite dimensional vector space over R. There
are many ways to put a norm on M m×n (R), here we consider some of the
most frequently used matrix norms.
Example 1.30. The Frobenius norm. If we regard the entries of a matrix
A = (aij ) as coordinates for a point in Euclidean space Rmn then we arrive
at
kAkF =
m X
n
X
a2ij
! 12
=
i=1 j=1
p
trace(At A)
We can regard an (m × n)-matrix as a continuous linear operator from Rn
to Rm . In the following three cases we use the operator norm on L(Rn , Rm )
while varying the norms on Rn and Rm .
Example 1.31. The 1-norm.
kAk1 =
sup kA(x)k1
kxk1 =1
=
m
X
max
1≤j≤n
|aij |
i=1
i.e. the maximum of the absolute column sums.
Example 1.32. The ∞-norm.
kAk∞ =
=
sup kA(x)k∞
kxk∞ =1
max
1≤i≤m
n
X
j=1
i.e. the maximum of the absolute row sums.
|aij |
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Example 1.33. The spectral norm (or 2-norm).
kAk2 =
sup kA(x)k2
kxk2 =1
=
p
largest eigenvalue of At A
Each of the above matrix norms is submultiplicative in the sense that
kABk ≤ kAk kBk
for all matrices A and B whenever the product AB is defined.