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:= max |Ax|2 = max xT (AT A)x = »max (AT A) = (AT A) .
A 2
|x|2 =1 |x|2 =1
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

= |»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
As a further example, the maximum row sum A ∞ is the matrix norm
induced by the maximum norm |x|∞ .
The number
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

(I ’ A) Aj ,

(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:
« 

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

It satis¬es the strict column sum criterion if the following relation holds:
|aij | < |ajj | for all j = 1, . . . , n .
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
|aij | ¤ |aii | holds for all i = 1, . . . , n

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
aii ≥ 0 and aij ¤ 0 (i = j)
are valid. An L0 -matrix is called an L-matrix if all diagonal entries are
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
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
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.
In this book, the ¬nite-dimensional vector space R is used in two as-

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 ⇐’
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)

slope: L
admissible region for f(y)

slope: -L


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)
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)

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 .

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

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
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:
|u| dx p ∈ [1, ∞) .
u := ,

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 ,

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

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

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