Appel W. Mathematics for Physics and Physicists

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Normed vector spaces 577

A.2

Normed vector spaces

A.2.a Norms, seminorms

In this section, K is either R or C, and E is a K-vector space, not necessarily

of ﬁnite dimension.

DEFINITION A.17 (Norm) Let E be a K-vector space. A norm on E is a map

N : E → R which satisﬁes the following properties, for all x, y ∈ E and

λ ∈ K:

N1 : we have N (x) ¾ 0;

N2 : we have



N(x) = 0 ⇐⇒ x = 0



;

N3 : we have N (λ x) = |λ| N(x);

N4 : we have N (x + y) ¶ N (x) + N ( y) (triangular inequality)

If is customary to write kxk for the norm of an element x, and to denote by

k·k the norm map it self.

A normed vector space is a pair (E, N ) where E is a vector space and N

is a norm on E.

DEFINITION A.18 (Seminorm) Let E be a K-vector space. A seminorm on E

is a map N : E → R satisfying properties N1, N3, and N4 a bove. Note that

N3 still implies N (0) = 0.

Example A.19 Let a, b ∈ R, a < b. Let C ([a, b]) be the vector sp ace of continuous functions

deﬁned on [a, b] , with values in K = R or C. For f ∈ C ([a, b]), le t

kf k



f (x)



dx norm of convergence in mean,

kf k



f (x)



dx norm of convergence in mean-square,

kf k

∞

= sup

[a,b]

|f | norme of uniform convergence.

Then k·k

, k·k

and k·k

∞

are norms on C ( [a, b]).

On the space of piecewise continuous functions, k·k

and k·k

are seminorms (a function which

is zero almost ever ywhere satisﬁes kf k

= kf k

= 0 but is not necessarily zero everywhere)

and k·k

∞

is a norm.

If a, b ∈ R, it i s necessary to restrict attention to integrable (resp. square integrable, resp.

bounded conti nuous) functi o ns in order that k·k

, k·k

, or k·k

∞

be deﬁned; then they are

norms.

578 Topology and normed vector spaces

DEFINITION A.20 (Distance) Let



E, k·k



be a normed vector space. The

distance associated to the norm k·k is the map

d : E

−→ R,

(x, y) 7−→ d(x, y)

def

= kx − yk.

Let A ⊂ E be a non-empty subset of E and let x ∈ E. The distance from x

to A is th e real number

d(x; A)

def

= inf

y∈A

kx − yk.

A.2.b Balls and topology associated to the distance

The distance associated to a norm g ives a normed vector space the structure of

a metric space. There is then a natural topology associated with this metric; it

is of course a (very) special case of the general topological notions described

in Section A.1.

DEFINITION A.21 Let (E, k·k) be a normed vector space. Let a ∈ E and let

r ∈ R

. The open ba ll centered at a with radius r (for the norm k·k) is the

set deﬁned by

B(a ; r)

def



x ∈ E ; kx −ak < r



The closed ball centered at a with radius r is the set deﬁned by

B(a ; r)

def



x ∈ E ; kx −ak ¶ r



DEFINITION A.22 (Open set) A subset A ⊂ E is open in E if and only if,

for any x ∈ A, there exists a ball centered at x with positive radius r > 0

contained in A:

∀x ∈ A ∃r > 0 B(x ; r) ⊂ A.

DEFINITION A.23 (Closed) A subset A ⊂ E is closed in E if and only if the

complement of A in E, denoted ∁

A, is open.

Example A.24 A well-known French saying is that “Il faut qu’une porte soit ouverte ou fermée”

(Musset; “A door must be ei ther open or closed”); this is quite true in its way, but not in the

setting of topologic al space s and normed vector spaces in particular. For instance, in R

, the

set ]0, 1[ ×{0} is neither open nor closed. Similarly, in R, Q is neither open nor closed. More

surprising maybe, R (in R) is both open and closed.

DEFINITION A.25 (Interior) L et A be a subset of E. A point a ∈ E is in the

interior of A if there exists r > 0 such tha t the open ball centered at a with

radius r is contained in A:

∃r > 0 B(a ; r) ⊂ A.

The interior of A, denoted

◦

A, is the set of all interior points of A.

Normed vector spaces 579

DEFINITION A.26 (Closure) A point a ∈ E is in the closure of A if, for

any posit ive r > 0, the open ball centered at a with radius r has non-empty

intersection w ith A:

∀r > 0 B(a ; r) ∩ A 6= ∅.

The closure of A, denoted A, is the set of all a satisfying this property.

PROPOSITION A.27 For any A ⊂ E, we have

◦

A ⊂ A ⊂ A.

Example A.28 Let A be a non-empty subset of R which h as an upper bound. Then sup A ∈ A.

PROPOSITION A.29 A subset A ⊂ E is closed if and only if A is equal to its

closure: A = A. A subset B ⊂ E is open if and only if B is equal to its interior:

◦

B = B.

The deﬁnition of a dense set may then be rephrased in this manner:

DEFINITION A.30 A subset A ⊂ E is dense in E if and only if the closure

of A is the whole space E.

Example A.31 The set Q of rationals i s dense in R; the set of invertible matrices GL

(K) is

dense is the space M

(K) of all matrices. The set of diagonalizable matrices is dense in M

(C).

THEOREM A.32 Let E and F be normed vector spaces, D ⊂ E a dense subset of E,

and f : E → F a continuous map. If f (x) = 0 for all x ∈ D, then f is zero on E.

Example A.33 Here is an application: for any matrix A ∈ M

(C), we have det(exp A) =

exp(tr A). This property i s obvious in the case of a diagonalizable matrix: indeed, assume that

A = P diag(λ

, . . . , λ

−1

(with P ∈ GL

(C)); then we have

exp A = P







exp(λ

)

exp(λ

)







−1

and

i=1

exp(λ

) = exp



i=1



Since the set of diagonalizable matrices is dense in M

(C), and since the determinant, th e trace

and the exponential function for matrices are continuous (the last being not quite obvious), it

follows that the identity is also valid for an arbitrary A ∈ M

(C).

Example A.34 Let f : [ 0 , 1 ] → R be a continuous function such that



x + y



f (x) + f ( y)

for all (x, y) ∈ R

. We want to prove that f is afﬁne, i.e., there exist a, b ∈ R such that

f (x) = ax + b for all x ∈ [0, 1]. For thi s purpose, let b = f (0) and a = f (1) − f (0), and let

g : x 7→ f (x) − (ax + b). The function g is then continuous, and satisﬁes g(0) = g(1) = 0.

Moreover, g satisﬁes the same relation



x + y



g(x) + g( y)

580 Topology and normed vector spaces

Edmund Geo rg Hermann Landau (1877—1938), German mathe-

matician, was famous in partic u lar for his many works of analytic

number theory (in partic u lar concerning the Riemann zeta func-

tion) and his treatise on number theory. From his youngest days,

he was also interested i n math ematical problems and games, and

had published two books of chess problems even before ﬁnishing

his thesi s. He taught at the University of Berlin from 1899 to 191 9,

then at the University of Göttingen (replacing Mi nkowski) until

he was forced to resign by the Nazis in 1934 (t he SS mathemati-

cian Teichmüller organized a boycott of his classes).

as before. From this we deduce immediately that g(

) = 0, then, repeating the process, that

) = g(

) = 0,

and by an obvious inducti o n we obtain





= 0 for all n ∈ N

∗

and all k ∈ [[0, 2

]].

Since the set E =



k ·2

−n

; n ∈ N

∗

, k ∈ [[0, 2

]]



is dense in [0, 1], and since g is continuous,

we can conclude that g = 0.

THEOREM A.35 (Properties of open sets) The union of any family of open sets is

an open set.

The intersection of ﬁnitely many open sets is an open set.

Example A.36

n∈N



, 1



= ]0, 1[ is open in R.

Counterexample A.37 The intersection of inﬁnitely many open sets has no reason to be open:

for instance,

n∈N



0, 1 +



= ]0, 1] is not open in R.

THEOREM A.38 (Properties of closed sets) The union of ﬁnitely many closed

sets is a closed set.

The intersection of any family of closed sets is a c losed set.

Counterexample A.39 As before, the union of inﬁnitely many closed sets has no reason to be

closed: for instance,

n∈N



0, 1 −



= [0, 1[ is not closed.

A.2.c Comparison of sequences

DEFINITION A.40 Let E be a normed vector space.

A sequence ( u

)

n∈N

with values in E is dominated by a sequence (α

)

n∈N

of non-negative real numbers, denoted u

= O(α

) and pronounced “u

big-Oh of α

,” if there exist a real number A and an integer N such that

k ¶ A |α

| for all n ¾ N .

Normed vector spaces 581

The Czech priest Bernhard Bolzano (1781—1848), was born and

died in Prague, in a German-speaking family of Italian origin. He

studied mathematics during his free time. Bec au se of his criticism

of the Austro-Hungarian Empire, he was forbidden to publish

his work and was closely watched. He sought to introduce rigor

in the foundations of mathematics: functi on theory, set theory,

logic. His impressive works (on cardinal theory, continuity, real

analysis) were in advance on their time and were neg lected during

his lifetime (his discovery of the Cauchy criterion for convergence

of sequences was not known). Finally, as a philosopher of science,

he exerted an important inﬂuence (notably on Husserl), and he

founded epistemology on formal logic.

The s equence ( u

) is negligible compared to (α

)

n∈N

if, for any positive

real number ǫ > 0, there exists an index N ∈ N, depending on ǫ, such t hat

k ¶ ǫ |α

| for all n ¾ N . This is denoted u

= o(α

), pronounced “u

little-Oh of α

.”

Two real- or complex-valued sequences u = ( u

)

n∈N

and v = (v

)

n∈N

are

equivalent, denoted u

∼ v

, if u

− v

= o(u

) or, equivalently (and it

shows t hat the relation “∼” is an equivalence relation), if u

−v

= o(v

The notation above is due to E. Landau.

A.2.d Bolzano-Weierstrass theorems

THEOREM A.41 (Bolzano-Weierstrass) Any bounded complex- or real-valued se-

quence has a convergent subsequence.

THEOREM A.42 (Bolzano-Weierstrass) Any subset of a ﬁnite-dimensional normed

vector space which is bounded and closed is compact, and in particula r, any sequence

with values in such a subset has a convergent subsequence.

A.2.e Comparison of norms

THEOREM A.43 Let E be a vector spa ce and let N , N

′

be two norms on E. Then

any sequence converging to 0 in the sense of N also converges to 0 in the sense of N

′

if and only if there exists a real number α ¾ 0 such that

′

¶ αN,

that is, N

′

(x) ¶ αN (x) for all x ∈ E.

DEFINITION A.44 Two norms N and N

′

on E are equivalent if there exist

real numbers α, β ¾ 0 such that

αN ¶ N

′

¶ βN .

582 Topology and normed vector spaces

Karl Theodor Wilhelm Weierstrass (1815—1897), German mathemati-

cian (from Westphalia), a famous teacher, was the ﬁrst to give a con-

vincing construction of the set R of real numbers (such a construc-

tion was lacking in Bolzano’s work). He also c onstruc ted an example

of a continuous function on R which is nowhere differentiable (ig-

norant of Bolzano’s prior work). The famous theorem stating that

any c ontinuous function on an interval [a, b] can be uniformly ap-

proximated by polynomials is also due to him. Finally, wishing to

provide rigorous and unambiguous deﬁnitions of the concepts of

analysis, h e introduced the deﬁnition of continuity based on “ep-

silons and deltas,” which are the cause of such happy moments and

memories in the lives of students everywhere.

THEOREM A.45 If the norms N and N

′

are equivalent, then any sequence that

converges for N also converges for N

′

and conversely, and moreover, the limits are

equal.

Example A.46 Let E = K

. We can easily compare the norms N

, N

, and N

∞

deﬁned by

( x) =

|, N

( x) =





1/2

and N

∞

( x) = max |x

|. Indeed, we have

∞

¶ N

¶ n N

∞

¶ N

n N

∞

hence

¶ N

¶ nN

Thus, those three norms are equivalent.

The previous example illustrates an impor tant t heorem in ﬁnite-dimen-

sional vector spaces:

THEOREM A.47 (Equivalence of norms) Let E be a ﬁnite-dimensional vector

space. Then all norms on E are equivalent. In particular, all notions deﬁned in

terms of the topology associated to a norm on E are identical whatever the norm used

in the deﬁnition (open sets, closed sets, compact sets, convergence of sequences, limits of

sequences, boundedness,

Cauchy sequences, Lipschitz functions,..).

Counterexample A.48 This is false for inﬁnite-dimensional vector spaces. For instance, con-

sider the space E = C



[0, 1] , R



of real-valued continuous functions on [0, 1], with the two

norms

( f ) = sup

t∈[0,1]



f (t)



and N

( f ) =



f (t)



dt.

It is clear that N

¶ N

, so any sequence of functions that converges uniformly to 0 also

converges in mean to 0. However, the converse is false: for instance, let ( f

)

n∈N

be the sequence

of functions deﬁned by f

(x) = x

. Then ( f

)

n∈N

converges to 0 in means (N

( f

) = 1/(n+1)),

whereas kf

∞

= 1 for all n ∈ N.

Counterexample A.49 On R, t h e norms of convergence in mean, in quadratic mean, and of

uniform convergence, are pairwise non-equivalent. To see this, let Λ be the function deﬁned

by Λ(x) = 1 −|x| if x ∈ [−1, 1] and Λ(x) = 0 otherwise.

The property of being bounded; of course, a bound for a set may depend on the norm.

Exercise 583

• The sequence of co ntinuous functi o ns on R deﬁned by f

(x) = Λ (x/n) /n converges

uniformly and in quadratic mean to 0, but it does not converge in mean to 0.

• The sequence of continuous functions on R deﬁned by g

(x) =

n Λ(nx) converges

in mean to 0, but does not converge in quadratic mean, or uniformly.

• The se quence of continuous functions on R deﬁned by h

(x) = Λ (x/n ) /

n converges

uniformly to 0, but converges neither in mean nor in quadratic mean.

The reader will now easily ﬁnd examples illustrating the other cases, using the same ideas.

Another fundamental result speciﬁc to ﬁ nite-dimensional vector spaces is

the following:

THEOREM A.50 (Finite-dimensional normed vector spaces are complete)

The vector spaces R, C, and more generally a ll ﬁnite-dimensional v ector spaces are

complete, that is, any Ca uchy sequence in a ﬁnite-dimensional vector space is convergent.

A.2.f Norm of a linear map

The continuity of a linear map between normed vector spaces is characterized

by the following theorem:

THEOREM A.51 Let E and F be normed vector spaces and let f ∈ L(E, F ) be

a linear map from E to F . Then f is continuous if and only if there exists a real

number k ∈ R such that kf (x)k

¶ k kxk

for all x ∈ E.

If f ∈ L(E, F ) is continuous, the real number deﬁned by

|||f ||| = s up

kxk



f (x)



= sup

x∈E\{0}



f (x)



kxk

is called the norm of f . Hence, by deﬁnition, we have



f (x)



¶ |||f |||·kxk

for x ∈ E.

The map f 7→ |||f ||| is a norm on the vector space of continuous linear maps f rom

E to F .

EXERCISE

 Exercise A.1 Let



E, k·k



be a normed vector space and let L (E) denote the vector space

of continuous maps from E to itself with the associated norm as above. Assume that (E, k·k)

is complete. Prove that



L (E) , |||·|||



is also complete.

584 Topology and normed vector spaces

SOLUTION

 Solution of exercise A.1. Let (u

)

n∈N

be a Cauchy sequence in L (E). For all x ∈ E, we

have



(x) − u

(x)



¶ kxk·



− u



and this proves that the sequence



(x)



is a Cauchy sequence in E. By assumption, it

converges; we then let

u(x) = li m

n→∞

(x) ∀x ∈ E.

This deﬁnes the map u : E → E, which a moment’s thought shows to be li near. Obviously, we

want to prove that u is continuous, and is the limit of (u

)

n∈N

i) The map u is in L (E).

By the Cauchy proper ty, there exists N ∈ N such that



− u



¶ 1 for all p, q ¾ N .

 Let x ∈ E be such that kxk = 1 .

For all p, q ¾ N, we have



(x) − u

(x)



¶ 1. Letting q go to +∞, we obtain



(x) − u(x)



¶ 1 for all p ¾ N . Hence,



u(x)



u(x) −u

(x)



(x)



¶ 1 + |||u

|||. 

This proves that

sup

kxk=1



u(x)



¶ 1 + |||u

|||

and i n particular this is ﬁnite, which means that u is continuous.

ii) We have lim

n→∞

= u in the sense of the no rm |||·||| on L (E).

 Let ǫ > 0. There exists N ∈ N such that



− u



¶ ǫ for all p, q ¾ N . Let

x ∈ E be such that kxk = 1. Then

∀p, q ¾ N



(x) − u

(x)



¶ ǫ,

and letting q go to +∞ again, we obtain

∀p ¾ N



(x) − u(x)



¶ ǫ

(we use the fact that the norm on E is itself a continuous map). Since this holds for all

x ∈ E such that kxk = 1, we have proved that

∀p ¾ N



− u



¶ ǫ. 

And of course, since ǫ > 0 was arbitrary, this is precisely the deﬁnition of the fact that

)

n∈N

converges to u i n L (E).

Appendix

Elementary reminders of

diﬀerential calculus

B.1

Differential of a real-valu e d function

B.1.a Functions of one re al variable

Let f : R → R be a function which is differentiable at a point a ∈ R. By

deﬁnition, this means that the limit

ℓ

def

= lim

x→a

f (x) − f (a)

x −a

= lim

h→0

f (x + h) − f (x)

exists. This also means that there exists a unique real number q ∈ R such that

f (x + h) = f (x) + q · h + o

h→0

(h).

In fact, we have q = ℓ, and this is the derivative of f at a, that is,

′

(a) = q = ℓ.

An equivalent rephrasing is that there exists a unique linear map Θ ∈

L (R) such that

f (x + h) = f (x) + Θ.h + o

h→0

(h),

586 Elementary reminders of differential calculus

since a linear map of R into itself is simply multiplication by a constant

value. (We use the notation Θ.h for the value of Θ evaluated at h, instead

of the more usual Θ(h); this is customary in differential calculus.) This new

formulation has the advantage of generalizing easily to a function of many

variables, that is, to functions deﬁned on R

B.1.b Diffe r ential of a function f : RR

→ RR

Let f : R

→ R be a real-valued function of n real variables, or one “ vector”

variable. We want to generalize the notion of derivative of f , at a point

a ∈ R

. First, the i-th partial derivative of f at the point a ∈ R

is deﬁned

as the limit

∂ f

∂ x

( a)

def

= lim

h→0

f ( a + h e

) − f ( a)

(if it exists ), where e

is the i-th vector of the canonical basis of R

. A classical

result of differential calculus states that if all partial derivatives ∂ f /∂ x

exist

and are continuous functions of a, then we have

f ( a + h) = f ( a) +

∂ f

∂ x

( a) · h

+ o

h→0

(h). (b.1)

for h ∈ R

. Denote by dx

the linear form “i-th coordinate” on R

, that is,

the linear map associating dx

.h = h

to a vector h = (h

, . . . , h

). Then we

can st ate the following deﬁnition:

DEFINITION B.1 (Differential) Let f : R

→ R be a map which has continu-

ous partial derivatives at a ∈ R

. The differential of f at a, denoted d f

is the linear map d f

: R

→ R deﬁned by

d f

.h =

i=1

∂ f

∂ x

( a) · h

or equivalently d f

i=1

∂ f

∂ x

( a) dx

We can then write (b.1) as follows:

f ( a + h) = f ( a) + d f

.h + o

h→0

(h).

This turns out to be the right way of deﬁning differentiable functions: f

is differentiable at a if and only if there exists a linear map Θ : R

→ R

such that

f ( a + h) = f ( a) + Θ.h + o

h→0

(h),

and the linear map Θ is the same as the differential deﬁ ned above using

partial derivatives.