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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 1. Topological, metric, functional, and vector spaces
p
L
p
L
p
L
f
p
L
p
L
L
L
p
L
L
f
p
L
f
p
f
f
f
g
11
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Example 1.1.3 (
Let T be a vector space of the all integrable on some set X functions, then
by
⋅
p
L
Example 1.1.4 (
sup ( )
Such a norm is called the uniform norm.
Remark 1.1.4
A. Condition
In any of functional spaces ,1
B.
-norm in an infinite dimensional vector space at finite
) is a map
∞
T+→
-norm in an infinite dimensional vector space of numerical functions)
defined by
⎛⎞
∀≡
ff fxdx
∈
fT
in Examples 1.1.1 and 1.1.3 is needed to satisfy inequality (1.1.4),
1p ≥
p
L
≡ . (1.1.12)
∞
L
()
⎜⎟
∫
⎜⎟
⎝⎠
X
fx
xX
∈
1/
p
. (1.1.11)
,1pp∈≥
known also as Minkowski inequality. At 1p < condition (1.1.4) fails.
p
Lp≤≤∞, space of continuous functions is dense in the
)
-norm (denoted
corresponding
C.
The following embedding of spaces
-topology.
and at
the topology
qp>
Theorem 1.1.1 (Hölder’s inequality)
A.
Let
convex function of
If integrable function
B.
Let
C.
be an integrable function, then a set I of real p,
are finite, is either empty, or a closed interval. In the latter case
p
L
1/
as the starting point. In the latter case
1p =
P
L∈ and
L∈ , where 1 p≤≤∞, 1 q≤≤∞ and
takes place:
qp
,
Lqp⊂>
q
is stronger than topology
, (1.1.13)
induced in
1 p≤≤∞
q
.
, at which
log( )
-norms
f
L
.
has a finite support, then the interval I is either empty, or has
is the increasing function of p.
p
L
q
11
+=
, (1.1.14)
1
pq
p
is a
Chapter 1. Topological, metric, functional, and vector spaces
f
f
g
y
y
y
g
g
y
yy
g
12
____________________________________________________________________________
1
then
gL∈
and
Remark 1.1.5
Inequality (1.1.15) is known as Hölder’s inequality. At 2p = (and hence 2q = ) inequality
(1.1.15) is also called as Cauchy – Bounjakowsky inequality.
Definition 1.1.12 (Hilbert space)
Function
mapping real vector space T into , or into , if T is a complex vector space, is
called linear, if
gfg≤
1 pq
LLL
(1.1.15)
1.1.3. Hilbert spaces
,( )()()
ggg
∀+=+
x
,
xy
∀∀ =
xxx
T
∈
x
x
T
∈
cgc cg
or
c
∈
x
. (1.1.16)
() ()
Function satisfying conditions (1.1.16) is quite often called linear form.
Definition 1.1.13 (Bilinear form)
Function
mapping real vector space
respect to each of its arguments. Such a function is quite often called bilinear form.
Definition 1.1.14 (Symmetric bilinear form)
Bilinear form
is called symmetric, if
,(,)(,)
∀=
x
,
T
∈
xy
Definition 1.1.15 (Sesquilinear form)
Function
mapping complex vector space
with respect to the first argument at fixed second argument, and the complex-conjugate
into , is called bilinear, if it is linear with
TT
×
gg
x
TT×
x
into , is called sesquilinear, if it is linear
. (1.1.17)
Chapter 1. Topological, metric, functional, and vector spaces
g
g
yy
y
y
yy
y
13
____________________________________________________________________________
function
second argument at fixed first argument. Such a function is quite often called a sesquilinear
form.
Definition 1.1.16 (Hermitian form)
(this will be precisely defined later on in this chapter) is linear with respect to its
Bilinear form
is called Hermitian, if
Definition 1.1.17 (Scalar product)
Symmetric or Hermitian form (depending upon real or complex vector space is considered) is
called scalar product, if
Quite often scalar product is denoted by
Definition 1.1.18 (Hilbert space)
A normed space (usually Banach space) is called Hilbert space, if it is complete and its norm is
defined by a scalar product
(, ) (,)gg=x
∀>
xxx. (1.1.19)
,0
Tg∈≠
xx
x . (1.1.18)
(,) 0
,⋅⋅
or
,⋅≡ ⋅⋅
.
,⋅⋅
()
. (1.1.20)
Remark 1.1.6
A
(Euclidian space)
finite dimensional vector space equipped with the scalar product is called Euclidean space.
Proposition 1.1.3 (Cauchy –Schwartz inequality, known also as the Cauchy – Bunyakovsky –
Schwartz inequality)
,, ,,
∀≤ =
,
xy
x
T∈
x
xx
. (1.1.21)
x
Proof
The proof directly follows from inequality (1.1.19), yielding
,,0
∀∀−−≥
xy xyxy
,
∈
xy
ccc
or
c
∈
T
. (1.1.22)
Chapter 1. Topological, metric, functional, and vector spaces
yy
p
L
fg fxg
14
____________________________________________________________________________
Taking
xx
c =
,
, (1.1.23)
,
where we assumed that
substituting (1.1.23) into (1.1.22), we arrive at the desired inequality (1.1.21).
Remark 1.1.7
Banach space
corresponding Hilbert space, if scalar multiplication is defined by
Definition 1.1.19 (Dual topological space)
Let
ℑ be a topological space, then the dual space
(i.e. continuous linear functions defined on ℑ and taking values in
introduced in Example 1.1.3, at
,0≠yy
,()()
(otherwise inequality (1.1.21) becomes trivial), and
can be associated with the
2p =
. (1.1.24)
∫
X
xdx≡
1.1.4. Duality
′
is a space of all continuous linear forms
ℑ
).
Definition 1.1.20 (Weak topology)
It is possible to introduce the weakest topology in
remain continuous. Such a topology is denoted by
topology
Similarly, in the dual space
dual topology
Definition 1.1.21 (Mackey topology)
In the initial topological space
the linear forms from the dual space ′ℑ remain continuous. Such a topology is known as
Mackey topology, this is denoted by
in ℑ, since it is not stronger than the initial topology in ℑ.
′
(,)
σℑ ℑ
, in which elements of ℑ regarded as linear forms, are continuous.
, in which all the linear forms from
ℑ
′
(, )
σℑℑ , and it is called a weakened
′
(which can have no topology at all) it is possible to introduce a
ℑ
it is possible to introduce the strongest topology, in which all
ℑ
′
(, )
τℑℑ .
′
ℑ
Chapter 1. Topological, metric, functional, and vector spaces
p
L
L
E
p
L
p
L
E
p
L
E
p
L
p
f
p
p
15
____________________________________________________________________________
Remark 1.1.8
The initial topology T in ℑ satisfies the following condition:
σℑℑ ≤ ≤τℑℑ , (1.1.25)
′′
(, ) (, )T
where sign “ ≤ ” means a weaker topology.
Proposition 1.1.4
Let real p 1 p≤≤∞ and q 1 q≤≤∞ satisfy relation (1.1.14), then topological spaces
q
of numerical functions are dual spaces.
1.1.5. Sobolev functional spaces
Definition 1.1.22 (Space of locally integrable functions; Sobolev space)
Let
be a topological space, and
having continuous derivatives up to
1 p≤<∞ induced in
k
()
CE
, where
k
be a set of all real (or complex) valued functions
()
CE
-th order. This set is not complete in
k
is a space of all locally integrable in p-th power
loc
-topology
loc
and
functions defined on
integrable on any bounded subsets of
defined by introducing the following semi-norms:
where ( )
k
()
CE in topology defined by (1.1.26) is called Sobolev space and denoted by
Remark 1.1.9
Quite often in applications topological space E is compact; for example, it can be a ball in
or a closed interval in
condition of integrability, and the corresponding Sobolev space is denoted by
. The term locally integrable means that functions from
. On ()
p
ff
,
kloc
k
CE∈ . Even in this stronger topology the space ( )
: In such a case condition of local integrability can be substituted by
k
≡
∑
0
m
=
k
CE a stronger topology than
m
()
, (1.1.26)
loc
k
CE is not complete. Closer of
W .
,
kloc
.
W
k
loc
can be
loc
are
n
Chapter 1. Topological, metric, functional, and vector spaces
p
p
p
16
____________________________________________________________________________
Proposition 1.1.5 (Sobolev embedding theorem)
Let E be a compact in
&/ /
n
and k, m be natural numbers and
km knpmnq>−>−, (1.1.27)
then
and embedding is continuous (
W
WW⊆
k
-topology is stronger than
k
Remarks 1.1.10 (Rellich-Kondrashov theorem)
A.
Sobolev embedding theorem is also known as Rellich-Kondrashov theorem.
Conditions of the theorem remain valid if q =∞. In such a case for any natural n and real
B.
(
1 p<<∞
) the following embedding theorem takes place:
p
WCE⊆
k
provided
/
knpm−>. (1.1.30)
1,pq<<∞
p
q
(1.1.28)
m
q
).
W
m
m
, (1.1.29)
()
. If
Thus, at satisfying condition (1.1.30) functions from
-th order.
m
C.
There are generalizations of Sobolev spaces with fractional index
have continuous derivatives up to
W
k
; these generalizations
k
known also as Hörmander spaces will be considered in Chapter 2.
1.2. Equation Chapter 1 Section 2 (Real)
trigonometric, hyperbolic, and some other
functions and series
In this section we present only those properties of the corresponding (real) elementary
functions that will be needed for the further analyses. For references see Korn and Korn
(2000) and Titchmarsh (1976).
Chapter 1. Topological, metric, functional, and vector spaces
x
17
____________________________________________________________________________
1.2.1. Trigonometric functions
Definition 1.2.1 (Sine function; Cosine function))
Sine and cosine functions can be defined by the following equivalent equations.
1) These are functions defined by the following series:
357 21
sin( ) ... ( 1)
xx x x x
x
≡−+−+= −
1! 3! 5! 7! (2 1)!
246 2
x
cos( ) 1 ... ( 1)
≡− + − += −
xx x
2! 4! 6! (2 )!
∑
n
∞
=
1
∞
∑
=
n
n
0
n
1
−
n
−
n
−
n
. (1.2.2)
n
2) These functions are solutions of the following differential equation:
2
⎛⎞
d
1()0
fx
+=
⎜⎟
2
⎜⎟
dx
⎝⎠
. (1.2.3)
Definition 1.2.2 (Real analytic function)
A function is (real) analytic (in particular vicinity), if it can be expanded into a power series,
convergent in that vicinity.
Proposition 1.2.1
Power series in the right-hand sides of (1.2.1), (1.2.2) converge everywhere in
ensuring both sine and cosine to be (real) analytic functions
(1.2.1)
(,)−∞ ∞
,
Proof
Proof of the proposition flows out directly from expressions (1.2.1), (1.2.2), and Stirling’s
estimate for the factorial; see:
!2explog1,nnnn n≈π − →∞. (1.2.4)
()
()
Combining (1.2.1), (1.2.2), (1.2.4) yields for sine and cosine functions:
k
exp log 2
kk
−−
()
where parameter
x
!
k
in (1.2.5) corresponds to
k
()
∼
2
π
k
,
k
21n −
, (1.2.5)
→∞
for sine and 2n for cosine function.
Asymptotic estimate (1.2.5) ensures convergence of the regarded series.
Chapter 1. Topological, metric, functional, and vector spaces
x
x
x
xxx
18
____________________________________________________________________________
Corollary
Any power series
k
∑
k
with coefficients
satisfying asymptotic estimate
a
k
(1.2.6)
a
k
!
k
ao kkk k→∞∼ , (1.2.7)
exp( log ) ,
()
k
defines a real analytic function in
denotes that sequence
increases weaker at
a
k
(,)−∞ ∞
. The symbol o, known as the small Landau symbol,
k →∞
than the expression in the right-hand
side of (1.2.7).
Proof
Proof flows out from estimate (1.2.5).
Remark 1.2.1
From definitions (1.2.1) – (1.2.3) it can be difficult, if possible, to deduce that both sine and
cosine are periodic functions.
We shall also need some other trigonometric functions, which definitions are given below.
Definition 1.2.3 (tangent function; Cotangent function)
Tangent and cotangent functions can be defined by the following equivalent equations.
1) These functions are expressed in terms of ratios of sine and cosine functions:
tan( )
sin( )
x
=
(1.2.8)
cos( )
cot( )
cos( )
= (1.2.9)
sin( )
2) These functions are expressed in terms of power series (convergent at
0 x<<π
for cotangent):
tan( ) ( 1)
22
∞
xx
=−
∑
=
1
k
kk
2(2 1)
−−
121
kk
(2 )!
B
−
2
k
(1.2.10)
k
π
< for tangent and
x
2
Chapter 1. Topological, metric, functional, and vector spaces
x
x
xxx
x
A
A
A
A
A
19
____________________________________________________________________________
2
∞
cot( ) ( 1)
12
=−
∑
xk
1
k
=
k
B
kk
2
2
k
, (1.2.11)
x
(2 )!
where
B are Bernoulli numbers. These numbers can be calculated by the following recurrent
2k
formula:
kk k
1, 1 ... 0
=+ + ++ =
BBBB
012
⎛⎞ ⎛⎞ ⎛ ⎞
⎜⎟ ⎜⎟ ⎜ ⎟
12 1
⎝⎠ ⎝⎠ ⎝ ⎠
k
k
−
. (1.2.12)
3) These functions are solutions of the following (nonlinear) differential equations:
2
⎛⎞
dd
tan( ) tan( ) 0
xx
−=
⎜⎟
2
⎜⎟
dx
⎝⎠
2
⎛⎞
dd
cot( ) cot( ) 0
+=
⎜⎟
2
⎜⎟
dx
⎝⎠
dx
xx
dx
, (1.2.13)
. (1.2.14)
Some basic properties of trigonometric functions:
22
sin ( ) cos ( ) 1xx+=, (1.2.15)
cos(2 ) cos ( ) sin ( ) 1 2sin ( ) 2 cos ( ) 1xxx x x=−=− = −, (1.2.17)
sin(2 ) 2 sin( )cos( )
22 2 2
xx=
, (1.2.16)
tan(2 )
cot(2 )
x
2tan( )
=
1tan()
−
2
cot ( ) 1
= , (1.2.19)
, (1.2.18)
2
x
−
2cot( )
BAB
sin( ) sin( ) 2sin cos
AB
±=
±
∓
, (1.2.20)
22
BAB
cos( ) cos( ) 2 cos cos
AB
+=
+−
, (1.2.21)
22
BAB
cos( ) cos( ) 2sin sin
AB
−=− , (1.2.22)
+−
22
±
tan( ) tan( )
±=
AB
sin( )
cos( )cos( )
B
, (1.2.23)
B
Chapter 1. Topological, metric, functional, and vector spaces
A
A
A
A
A
A
A
x
x
x
x
x
∑
20
____________________________________________________________________________
±
cot( ) cot( )
±=− , (1.2.24)
AB
sin( )
sin( )sin( )
B
B
sin( ) sin( )cos( ) sin( )cos( )
cos( ) cos( )cos( ) sin( )sin( )
cot( )
AB
BABBA±= ±
BABAB±= ∓
±=
tan( )
AB
cot( )cot( ) 1 sinh( ) cosh( )
±=
tan( ) tan( )
1tan()tan()
Bxx
±
∓
d
∓
dx
cot( ) cot( )
BA
±
B
B
, (1.2.25)
, (1.2.26)
, (1.2.27)
=
, (1.2.28)
dd
sin() cos sinh() cosh()()
==
xxx
, (1.2.29)
dx dx
d
cos( ) sin( )
=− , (1.2.30)
x
dx
d
tan( )
x
dx
d
cot( )
x
dx
=
cos ( )
=−
1
2
1
2
sin ( )
, (1.2.31)
. (1.2.32)
1.2.2. Exponential and hyperbolic functions
Definition 1.2.4 (Exponential function)
Exponential function can be defined by the following equivalent equations:
1) This a function is defined by the following series:
23
exp( ) 1 ...
≡+ + + + =
x
xx x
1! 2! 3! !
2) This is a solution of the following differential equation [1]:
d
⎛⎞
1()0
−=
⎜⎟
dx
⎝⎠
fx
(1.2.34)
n
∞
(1.2.33)
n
n
1
=
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