Chapter 1
Preliminary
Abstract
In this chapter we shall introduce and state some necessary
materials needed in the proof of our results, and shortly the basic results
which concerning the Banach spaces, the weak and weak star topologies, the
II space, Sobolev spaces and other theorems. The knowledge of all
this notations and results are important for our study.
1.1 Banach Spaces - Definition and Properties
We first review some basic facts from calculus in the most
important class of linear spaces "Banach spaces".
Definition 1.1.1 A Banach space is a complete normed linear space
X. Its dual space X' is the linear space of all continuous linear
functional f : X --> R.
Proposition 1.1.1 ([43])
X' equipped with the norm 11.11x,
defined by
II/11x, = sup flf(u)I : Ilull 1}, (1.1)
is also a Banach space.
We shall denote the value of f 2 X' at u 2 X by either
f(u) or (f, u)x,,x
·
Remark 1.1.1 ([43]) From X' we construct the bidual
or second dual X'' = (X')'. Furthermore, with
each u 2 X we can define cp(u) 2 X'' by cp(u)(f) = f(u), f 2
X', this satisfies clearly Ip(x)1 < Mull . Moreover, for each u 2
X there is an f 2 X' with f(u) = Mull and 11f11 = 1, so it follows
that Ip(x)1 = Mull .
Definition 1.1.2 Since cp is linear we see that
cp : X --> X'',
is a linear isometry of X onto a closed subspace of
X'', we denote this by
X c-- X''.
Definition 1.1.3 If cp ( in the above definition) is onto
X'' we say X is reflexive, X c---,'
X''.
Theorem 1.1.1 ([4], Theorem III.16)
Let X be Banach space. Then, X is reflexive, if and only if,
Bx = Ix 2 X :114 <11,
is compact with the weak topology a
(X, X') . (See the next subsection for the definition of a (X,
X'))
Definition 1.1.4 Let X be a Banach space, and let
(un)nEN be a sequence in X. Then un converges
strongly to u in X if and only if
lim
n-->o
Ilun -- ullx = 0,
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and this is denoted by un --p u, or lim
n-->o
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Un = U.
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1.1.1 The weak and weak star topologies
Let X be a Banach space and f E X'. Denote by
cpf : X --> R
x i--> Wf(x),
(1.2)
when f cover X', we obtain a family
(cpf)fcx, of applications to X in R.
Definition 1.1.5 The weak topology on X, denoted by a (X,
X') , is the weakest topology on X for which every
(pf)fcx, is continuous.
We will define the third topology on X', the weak star
topology, denoted by a (X', X) . For all
x E X. Denote by
cpx : X' --> R
f i- wx(f) = (f, x)x,,x, (1.3)
when x cover X, we obtain a family
(cpx)xcx, of applications to X' in
R.
Definition 1.1.6 The weak star topology on X' is the
weakest topology on X' for which every (4x)xcx, is
continuous.
Remark 1.1.2 ([4]) Since X C X'', it is clear that,
the weak star topology a (X', X) is weakest then the topology a
(X', X''), and this later is weakest then the strong
topology.
Definition 1.1.7 A sequence (un) in X is weakly
convergent to x if and only if
for every f E X', and this is denoted by un
--, u.
Remark 1.1.3 ([42], Remark 1.1.1)
1. If the weak limit exist, it is unique.
2. If un --> u E X (strongly), then un
--, u (weakly).
3. If dim X < +oo, then the weak convergent implies the
strong convergent.
Proposition 1.1.2 ([43])
On the compactness in the three topologies in the Banach space X
:
1- First, the unit ball
B Ix E X : 11x11 < 1}, (1.4)
in X is compact if and only if dim(X) < oc.
2- Second, the unit ball B' in X' (The
closed subspace of a product of compact spaces) is weakly compact in
X' if and only if X is reflexive.
3- Third, B' is always weakly star compact in the
weak star topology of X'.
Proposition 1.1.3 ([4], proposition III.12)
Let (fn) be a sequence in X'. We have:
[ ]
~
1. fm * f in a (X', X) [fn(x) !
f(x), Vx 2 X].
2. If fTh - f (strongly), then fTh - f, in a
(X', X''), If fm - f in a
(X',X''), then fm - f, in a
(X',X).
~
3. If fTh - f, in a (X',X), then
kfnk is bounded and kfk ~ liminf MfnM.
~
4. If fTh - f, in a (X', X) and x, -p x
(strongly) in X, then fn(xn) -p f(x).
~
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