A 2

|x|2 =1 |x|2 =1

(A3.10)

The matrix norm A 2 induced by the Euclidean vector norm is also called

the spectral norm. This term becomes understandable in the special case of

a symmetric matrix A. If »1 , . . . , »n denote the real eigenvalues of A, then

the matrix AT A = A2 has the eigenvalues »2 satisfying

i

= |»max (A)| .

A 2

For symmetric matrices, the spectral norm coincides with the spectral ra-

dius. Because of (A3.8), it is the smallest possible matrix norm in that

case.

As a further example, the maximum row sum A ∞ is the matrix norm

induced by the maximum norm |x|∞ .

The number

A’1

κ(A) := A

is called the condition number of the matrix A with respect to the matrix

norm under consideration.

398 A. Appendices

The following relation holds:

1 ¤ I = AA’1 ¤ A A’1 .

For | · | = | · |p , the condition number is also denoted by κp (A). If all

eigenvalues of A are real, the number

κ(A) := »max (A)/»min (A)

is called the spectral condition number. Hence, for a symmetric matrix A

the equality κ(A) = κ2 (A) is valid.

Occasionally, it is necessary to estimate small perturbations of nonsin-

gular matrices. For this purpose, the following result is useful (perturbation

lemma or Neumann™s lemma). Let A ∈ Rn,n satisfy A < 1 with respect

to an arbitrary, but ¬xed, matrix norm. Then the inverse of I ’ A exists

and can be represented as a convergent power series of the form

∞

’1

(I ’ A) Aj ,

=

j=0

with

1

(I ’ A)’1 ¤ . (A3.11)

1’ A

Special Matrices

The matrix A ∈ Rn,n is called an upper, respectively lower, triangular

matrix if its entries satisfy aij = 0 for i > j, respectively aij = 0 for i < j.

A matrix H ∈ Rn,n is called an (upper) Hessenberg matrix if it has the

following structure:

«

h11

—

¬ ·

¬ h21 . . . ·

¬ ·

¬ ·

.. ..

H := ¬ ·

. .

¬ ·

¬ ·

.. ..

¬ ·

. .

0 hnn’1 hnn

(that is, hij = 0 for i > j + 1).

The matrix A ∈ Rn,n satis¬es the strict row sum criterion (or is strictly

row diagonally dominant) if it satis¬es

n

|aij | < |aii | for all i = 1, . . . , n .

j=1

j=i

It satis¬es the strict column sum criterion if the following relation holds:

n

|aij | < |ajj | for all j = 1, . . . , n .

i=1

i=j

A.4. Some De¬nitions and Arguments of Linear Functional Analysis 399

The matrix A ∈ Rn,n satis¬es the weak row sum criterion (or is weakly row

diagonally dominant) if

n

|aij | ¤ |aii | holds for all i = 1, . . . , n

j=1

j=i

and the strict inequality “<” is valid for at least one number

i ∈ {1, . . . , n} .

The weak column sum criterion is de¬ned similarly.

The matrix A ∈ Rn,n is called reducible if there exist subsets N1 , N2 ‚

{1, . . . , n} with N1 © N2 = …, N1 = … = N2 , and N1 ∪ N2 = {1, . . . , n} such

that the following property is satis¬ed:

For all i ∈ N1 , j ∈ N2 : aij = 0 .

A matrix that is not reducible is called irreducible.

A matrix A ∈ Rn,n is called an L0 -matrix if for i, j ∈ {1, . . . , n} the

inequalities

aii ≥ 0 and aij ¤ 0 (i = j)

are valid. An L0 -matrix is called an L-matrix if all diagonal entries are

positive.

A matrix A ∈ Rn,n is called monotone (or of monotone type) if the

relation Ax ¤ Ay for two (otherwise arbitrary) elements x, y ∈ Rn implies

x ¤ y. Here the relation sign is to be understood componentwise.

A matrix of monotone type is invertible.

A matrix A ∈ Rn,n is a matrix of monotone type if it is invertible and

all entries of the inverse are nonnegative.

An important subclass of matrices of monotone type is formed by the

so-called M-matrices.

A monotone matrix A with aij ¤ 0 for i = j is called an M-matrix.

Let A ∈ Rn,n be a matrix with aij ¤ 0 for i = j and aii ≥ 0 (i, j ∈

{1, . . . , n}). In addition, let A satisfy one of the following conditions:

(i) A satis¬es the strict row sum criterion.

(ii) A satis¬es the weak row sum criterion and is irreducible.

Then A is an M-matrix.

A.4 Some De¬nitions and Arguments of Linear

Functional Analysis

Working with vector spaces whose elements are (classical or generalized)

functions, it is desirable to have a measure for the “length” or “magnitude”

of a function, and, as a consequence, for the distance of two functions.

400 A. Appendices

Let V be a real vector space (in short, an R vector space) and let ·

be a real-valued mapping · : V ’ R.

The pair (V, · ) is called a normed space (“V is endowed with the norm

· ”) if the following properties hold:

u ≥ 0 for all u ∈ V , u = 0 ” u = 0, (A4.1)

±u = |±| u for all ± ∈ R , u ∈ V , (A4.2)

u+v ¤ u + v for all u, v ∈ V . (A4.3)

The property (A4.1) is called de¬niteness; (A4.3) is called the triangle

inequality. If a mapping · : V ’ R satis¬es only (A4.2) and (A4.3), it

is called a seminorm. Due to (A4.2), we still have 0 = 0, but there may

exist elements u = 0 with u = 0.

A particularly interesting example of a norm can be obtained if the space

V is equipped with a so-called scalar product. This is a mapping ·, · :

V — V ’ R with the following properties:

(1) ·, · is a bilinear form, that is,

for all u, v1 , v2 ∈ V ,

u, v1 + v2 = u, v1 + u, v2

(A4.4)

for all u, v ∈ V, ± ∈ R ,

u, ±v = ± u, v

and an analogous relation is valid for the ¬rst argument.

(2) ·, · is symmetric, that is,

for all u, v ∈ V .

u, v = v, u (A4.5)

(3) ·, · is positive, that is,

u, u ≥ 0 for all u ∈ V . (A4.6)

(4) ·, · is de¬nite, that is,

u, u = 0 ” u = 0 . (A4.7)

A positive and de¬nite bilinear form is called positive de¬nite.

A scalar product ·, · de¬nes a norm on V in a natural way if we set

1/2

v := v, v . (A4.8)

In absence of the de¬niteness (A4.7), only a seminorm is induced.

A norm (or a seminorm) induced by a scalar product (respectively by a

symmetric and positive bilinear form) has some interesting properties. For

example, it satis¬es the Cauchy“Schwarz inequality, that is,

| u, v | ¤ u for all u, v ∈ V ,

v (A4.9)

and the parallelogram identity

+ u’v + v 2 ) for all u, v ∈ V .

2 2 2

u+v = 2( u (A4.10)

A.4. Linear Functional Analysis 401

Typical examples of normed spaces are the spaces Rn equipped with one

of the p -norms (for some ¬xed p ∈ [1, ∞]). In particular, the Euclidean

norm (A3.3) is induced by the Euclidean scalar product

(x, y) ’ x · y for all x, y ∈ Rn . (A4.11)

On the other hand, in¬nite-dimensional function spaces play an important

role (see Appendix A.5).

If a vector space V is equipped with a scalar product ·, · , then, in

analogy to Rn , an element u ∈ V is said to be orthogonal to v ∈ V if

u, v = 0 . (A4.12)

Given a normed space (V, · ), it is easy to de¬ne the concept of convergence

of a sequence (ui )i in V to u ∈ V :

ui ’ u for i ’ ∞ ⇐’ ui ’ u ’ 0 for i ’ ∞ . (A4.13)

Often, it is necessary to consider function spaces endowed with di¬erent

norms. In such situations, di¬erent kinds of convergence may occur. How-

ever, if the corresponding norms are equivalent, then there is no change

in the type of convergence. Two norms · 1 and · 2 in V are called

equivalent if there exist constants C1 , C2 > 0 such that

¤u ¤ C2 u for all u ∈ V .

C1 u (A4.14)

1 2 1

If there is only a one-sided inequality of the form

¤C u for all u ∈ V

u (A4.15)

2 1

with a constant C > 0, then the norm · 1 is called stronger than the

norm · 2 .

In a ¬nite-dimensional vector space, all norms are equivalent. Examples

can be found in Appendix A.3. In particular, it is important to observe that

the constants may depend on the dimension n of the ¬nite-dimensional

vector space. This observation also indicates that in the case of in¬nite-

dimensional vector spaces, the equivalence of two di¬erent norms cannot

be expected, in general.

As a consequence of (A4.14), two equivalent norms · 1 , · 2 in V yield

the same type of convergence:

ui ’ u w.r.t. · ” ui ’ u ’0

1 1

” ui ’ u ’0 ” ui ’ u w.r.t.

· 2.

2

(A4.16)

In this book, the ¬nite-dimensional vector space R is used in two as-

n

pects: For n = d, it is the basic space of independent variables, and for

n = M or n = m it represents the ¬nite-dimensional trial space. In the

¬rst case, the equivalence of all norms can be used in all estimates without

any side e¬ects, whereas in the second case the aim is to obtain uniform

402 A. Appendices

estimates with respect to all M and m, and so the dependence of the

equivalence constants on M and m has to be followed thoroughly.

Now we consider two normed spaces (V, · V ) and (W, · W ). A mapping

f : V ’ W is called continuous in v ∈ V if for all sequences (vi )i in V with

vi ’ v for i ’ ∞ we get

f (vi ) ’ f (v) for i ’ ∞.

Note that the ¬rst convergence is measured in · V and the second

one in · W . Hence a change of the norm may have an in¬‚uence on the

continuity. As in classical analysis, we can say that

f is continuous in all v ∈ V ⇐’

(A4.17)

f ’1 [G] is closed for each closed G ‚ W .

Here, a subset G ‚ W of a normed space W is called closed if for any

sequence (ui )i from G such that ui ’ u for i ’ ∞ the inclusion u ∈

G follows. Because of (A4.17), the closedness of a set can be veri¬ed by

showing that it is a continuous preimage of a closed set.

The concept of continuity is a qualitative relation between the preimage

and the image. A quantitative relation is given by the stronger notion of

Lipschitz continuity:

A mapping f : V ’ W is called Lipschitz continuous if there exists a

constant L > 0, the Lipschitz constant, such that

f (u) ’ f (v) ¤ L u’v for all u, v ∈ V . (A4.18)

W V

slope: L

admissible region for f(y)

f

slope: -L

x

Figure A.1. Lipschitz continuity (for V = W = R).

A Lipschitz continuous mapping with L < 1 is called contractive or a

contraction; cf. Figure A.1.

Most of the mappings used are linear; that is, they satisfy

f (u + v) = f (u) + f (v) ,

for all u, v ∈ V and » ∈ R . (A4.19)

f (»u) = »f (u) ,

For a linear mapping, the Lipschitz continuity is equivalent to the

boundedness; that is, there exists a constant C > 0 such that

¤C u for all u ∈ V .

f (u) (A4.20)

W V

A.4. Linear Functional Analysis 403

In fact, for a linear mapping f, the continuity at one point is equivalent

to (A4.20). Linear, continuous mappings acting from V to W are also

called (linear, continuous) operators and are denoted by capital letters,

for example S, T, . . . .

In the case V = W = Rn , the linear, continuous operators in Rn are

the mappings x ’ Ax de¬ned by matrices A ∈ Rn,n . Their boundedness,

for example with respect to · V = · W = · ∞ , is an immediate

consequence of the compatibility property of the · ∞ -norm. Moreover,

since all norms in Rn are equivalent, these mappings are bounded with

respect to any norms in Rn .

Similarly to (A4.20), a bilinear form f : V — V ’ R is continuous if it is

bounded, that is, if there exists a constant C > 0 such that

|f (u, v)| ¤ C u for all u, v ∈ V .

v (A4.21)

V V

In particular, due to (A4.9) any scalar product is continuous with respect

to the induced norm of V ; that is,

ui ’ u , vi ’ v ’ ui , vi ’ u, v . (A4.22)

Now let (V, · V ) be a normed space and W a subspace that is (addi-

tionally to · V ) endowed with the norm · W . The embedding from

(W, · W ) to (V, · V ) , i.e., the linear mapping that assigns any element

of W to itself but considered as an element of V, is continuous i¬ the norm

· W is stronger than the norm · V (cf. (A4.15)).

The collection of linear, continuous operators from (V, · V ) to (W, · W )

forms an R vector space with the following (argumentwise) operations:

for all u ∈ V ,

(T + S)(u) := T (u) + S(u)

for all u ∈ V ,

(»T )(u) := »T (u)

for all operators T, S and » ∈ R. This space is denoted by

L[V, W ] . (A4.23)

In the special case W = R, the corresponding operators are called linear,

continuous functionals, and the notation

V := L[V, R] (A4.24)

is used. The R vector space L[V, W ] can be equipped with a norm, the

so-called operator norm, by

u∈V , u ¤1 for T ∈ L[V, W ] . (A4.25)

T := sup T (u) W V

Here T is the smallest constant such that (A4.20) holds. Speci¬cally, for

a functional f ∈ V , we have that

f = sup |f (u)| ¤1 .

u V

404 A. Appendices

For example, in the case V = W = Rn and u V = u W , the norm of a

linear, bounded operator that is represented by a matrix A ∈ Rn,n coincides

with the corresponding induced matrix norm (cf. Appendix A.3).

Let (V, · V ) be a normed space. A sequence (ui )i in V is called a Cauchy

sequence if for any µ > 0 there exists a number n0 ∈ N such that

ui ’ uj ¤µ for all i, j ∈ N with i, j ≥ n0 .

V

The space V is called complete or a Banach space if for any Cauchy sequence

(ui )i in V there exists an element u ∈ V such that ui ’ u for i ’ ∞. If

the norm · V of a Banach space V is induced by a scalar product, then

V is called a Hilbert space.

A subspace W of a Banach space is complete i¬ it is closed. A basic

problem in the variational treatment of boundary value problems consists

in the fact that the space of continuous functions (cf. the preliminary de¬-

nition (2.7)), which is required to be taken as a basis, is not complete with

respect to the norm ( · l , l = 0 or l = 1). However, if in addition to the

normed space (W, · ), a larger space V is given that is complete with

respect to the norm · , then that space or the closure

W := W (A4.26)

(as the smallest Banach space containing W ) can be used. Such a com-

pletion can be introduced for any normed space in an abstract way. The

problem is that the “nature” of the limiting elements remains vague.

If the relation (A4.26) is valid for some normed space W, then W is

called dense in W . In fact, given W, all “essential” elements of W are

already captured. For example, if T is a linear, continuous operator T from

(W , · ) to another normed space, then the identity

T (u) = 0 for all u ∈ W (A4.27)

is su¬cient for

for all u ∈ W .

T (u) = 0 (A4.28)

The space of linear, bounded operators is complete if the image space is

complete. In particular, the space V of linear, bounded functionals on the

normed space V is always complete.

A.5 Function Spaces

In this section G ‚ Rd denotes a bounded domain.

The function space C(G) contains all (real-valued) functions de¬ned on

G that are continuous in G. By C l (G), l ∈ N, the set of l-times continuously

di¬erentiable functions on G is denoted. Usually, for the sake of consistency,

∞

the conventions C 0 (G) := C(G) and C ∞ (G) := l=0 C l (G) are used.

A.5. Function Spaces 405

Functions from C l (G), l ∈ N0 , and C ∞ (G) need not be bounded, as for

d = 1 the example f (x) := x’1 , x ∈ (0, 1) shows.

To overcome this di¬culty, further spaces of continuous functions are

introduced. The space C(G) contains all bounded and uniformly contin-

uous functions on G, whereas C l (G), l ∈ N, consists of functions with

bounded and uniformly continuous derivatives up to order l on G. Here the

∞

conventions C 0 (G) := C(G) and C ∞ (G) := l=0 C l (G) are used, too.

The space C0 (G), respectively C0 (G), l ∈ N, denotes the set of all those

l

continuous, respectively l-times continuously di¬erentiable, functions, the

supports of which are contained in G. Often this set is called the set of

functions with compact support in G. Since G is bounded, this means that

0

the supports do not intersect boundary points of G. We also set C0 (G) :=

C0 (G) and C0 (G) := C0 (G) © C ∞ (G).

∞

The linear space Lp (G), p ∈ [1, ∞), contains all Lebesgue measurable

functions de¬ned on G whose pth power of their absolute value is Lebesgue

integrable on G. The norm in Lp (G) is de¬ned as follows:

1/p

|u| dx p ∈ [1, ∞) .

p

u := ,

0,p,G

G

In the case p = 2, the speci¬cation of p is frequently omitted; that is,

u 0,G = u 0,2,G. The L2 (G)-scalar product

u, v ∈ L2 (G) ,

u, v := uv dx ,

0,G

G

induces the L2 (G)-norm by setting u := u, u 0,G .

0,G

The space L∞ (G) contains all measurable, essentially bounded functions