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**CHAPTER ONE**

FOUNDATION

1.1 Basic notions and results from Functional

Analysis

The purpose of this section is to refresh our minds on some basic fundamentals

required to facilitate a smooth understanding of the study of compact linear

operators and its applications.

Denition 1.1.1. A non-negative function jj jj on a vector space X is called

a norm on X if and only if

i) jjxjj 0 for every x 2 X (Positivity).

ii) jjxjj = 0 if and only if x = 0 (Nondegeneracy).

iii) jj(x)jj = jjjjxjj for every x 2 X (Homogeneity).

iv) jjx + yjj jjxjj + jjyjj for every x; y 2 X (subadditivity).

A vector space X with a norm jj jj is denoted by (X; jj jj) and is called a

normed linear space (or just a normed space).

A sequence (xn)n2N of elements in a normed linear space X is called Cauchy

if 8 > 0; 9N 2 N, such that for m; n 2 N, jjxm xnjj , m; n N

A Banach space is a normed linear space (X; jj jj) that is complete in the

canonical metric dened by (x; y) = jjx yjj for x; y 2 X i.e every Cauchy

sequence in X for the metric converges to some point in X.

1

Remark. Every normed linear space has a completion [3]

Denition 1.1.2. Let X and Y be K-linear spaces (K = R or K = C). A

map T : X ! Y is called linear if

T(f + g) = T(f) + T(g) for all f; g 2 X; and all ; 2 K:

More generally, we can consider linear maps dened on a sub-space D(T)

of X and with values in Y . The Subspace D(T) X is called the domain of

T. We denote the image (or Range) of T by R(T) and is dened by

R(T) :=

y 2 Y : y = Tx for some x 2 D(T)

:

We dene the Kernel (or Null space) of T denoted by N(T) to be the subspace

of X dened by

N(T) := fx 2 D(T) : Tx = 0g

T is said to be injective (or one to one ) if N(T) = f0g.

T is said to be surjective (or onto) if R(T) = Y .

Denition 1.1.3. A mapping T : X ! Y is called a continuous linear

operator, if T is linear and is continuous at each point in a 2 X, that is

lim

x!a

Tx = Ta for all a 2 X :

The space of continuous linear operators from a Banach space X into a

Banach Space Y is denoted by B(X; Y ).

Proposition 1.1.4. [2]

Let X, Y be two normed spaces and BX be the closed unit ball of X. Let

T : X ! Y be a linear map. Then the following are equivalent :

i) T is a bounded linear operator, i.e there exists a constant M > 0 such that

jjTxjjY MjjxjjX for all x 2 X.

ii) T is continuous.

iii) T is continuous at the origin (in the sense that if fxng is a sequence in

X. such that xn ! 0 as n ! 0 , then Txn ! 0 in Y as n ! 1).

iv) T is Lipschitz.

v) T(BX) is bounded (i.e there exists a constant C > 0 such that jjTxjj C

for all x 2 BX).

2

Furthermore, B(X; Y ) becomes naturally endowed with the operator norm,

jjTjj := sup

x2X;x6=0

jjTxjjY

jjxjjX

= sup

jjxjjX1

jjTxjjY ;

and if Y is a Banach space then so is B(X; Y ).

We also recall that every linear operator of a nite dimensional space is bounded.

Proposition 1.1.5. Every bounded linear map between normed linear space

has a unique extension between their completions [4].

Theorem 1.1.6. Given a continuous complex function G on [a; b] [a; b],

let T be dened on X = C[a; b] at each f by

(Tf)(x) =

Z b

a

G(x; y)f(y)dy for all x 2 [a; b]:

T is called an integral operator with the kernel function G. Then T 2 B(X; Y )

and

jjTjj = max

axb

Z b

a

jG(x; y)jdy

Proof.

Firstly, we show that T is well dened, i.e., for every f 2 C[a; b], Tf 2 C[a; b].

Let x1; x2 2 [a; b], then

jTf(x1) Tf(x2)j

Z b

a

jG(x1; t) G(x2; t)jjf(t)jdt

max

t2[a;b]

jG(x1; t) G(x2; t)jjjfjj1(b a)

Since G is continuous on the compact set C([a; b] [a; b]) by assumption, it

follows that G is uniformly continuous and so we deduce from the above inequality

that Tf is uniformly continuous. Thus Tf 2 C[a; b].

The linearity of T follows from the linearity of the integral.

Now we investigate the boundedness of T. We have,

j(Tf)(x)j

Z b

a

jG(x; y)jdy (1.1.1)

and so,

jjTfjj1 = max

axb

j(Tf)(x)j max

axb

Z b

a

jG(x; y)jdyjjfjj1

showing that T is bounded and

kTk max

axb

Z b

a

jG(x; y)jdy :

3

Hence

kTk M; where M :=

Z b

a

jG(x; y)jdy:

Now we dene,

S(x) :=

Z b

a

jG(x; y)jdy:

S is continuous on [a; b] and therefore it attains its maximum at some x0 2

[a; b]:

Let g(y) :=

8<

:

jG(x0;y)j

G(x0;y) if G(x0; y) 6= 0

0 otherwise

Then the function g is bounded and Lebesgue measurable and so belongs to

L1([a; b]). It follows from the fact that C[a; b] is dense in L1([a; b]) that there

exists a sequence (gn)n of elements of C[a; b] such that kgnk = max

y2[0;1]

jgn(y)j 1

and (gn) converges in L1([a; b]) to g. Hence we have ,

kTk = sup

jjfjj1

kTfk kTgnk = max

axb

j(Tgn)xj (Tgn)(x0)

) kTk (Tgn)(x0) =

Z b

a

K(x0; y)gn(y)dy !

Z b

a

K(x0; y)g(y)dy

therefore, kTk

Z b

a

K(x0; y)g(y)dy =

Z b

a

jK(x0; y)jdy = M

It follows that jjTjj M: Hence,

kTk = M

Corollary 1.1.7. Given a continuous complex valued function G on

[0; 1] [0; 1], let T be dened on X = C[0; 1] by

(Tf)(x) =

Z 1

0

G(x; y)f(y)dy ; 8f 2 C[a; b]:

Then T is a bounded linear map with

jjTjj = max

0x1

Z 1

0

jG(x; y)jdy :

Denition 1.1.8. A map T dened from a Banach space X into a Banach

space Y is called closed if its graph

G(T) = f(x; y) : x 2 D(T)g

is closed in X Y . In other words, T is closed if whenever xn ! x and

Txn ! y, we have x 2 D(T) and Tx = y

4

Remark. Every T 2 B(X; Y ) is closed

Theorem 1.1.9. Closed Graph Theorem

A closed linear map which maps a Banach space into a Banach space is con-

tinuous

Denition 1.1.10. A map T 2 B(X; Y ) is invertible if there is a bounded

linear map T1 2 B(Y;X) such that T1T = IX (the identity of X) and

TT1 = IY (the identity operator of Y )

Corollary 1.1.11. Every continuous bijection between Banach spaces has a

continuous inverse.

Proof.

Let T 2 B(X; Y ). Then G(T) = f(x; Tx) : x 2 Xg is closed in X Y ,

G(T1) = f(Tx; x) : x 2 Xg is closed in Y X

T1 is a closed linear operator mapping Y into X.Therefore T1 is bounded

by the closed graph theorem.

Denition 1.1.12. Let A be a subset of a Banach space X. A is said to

be precompact if any sequence of A has a Cauchy subsequence. A is totally

bounded if for every > 0, there exists a nite cover for A of open balls of

same radius

Denition 1.1.13. A is said to be compact if any sequence of A has a subse-

quence that converges to some point of A.

Remark. A set is said to be precompact in a Banach X if and only if its closure

is compact in X.

Proposition 1.1.14. Let A be a subset of a Banach space X. A is said to be

precompact if and only if it is totally bounded.

Proof.

Assume that A is a precompact subset of X, we show that A is totally

bounded.

Let > 0 and x 2 A

x 2 A ) B(x; )

\

A 6= ;

) 9 ax 2 A : ax 2 B(x; )

) 9 ax 2 A : d(ax; x) <

) 9 ax 2 A : x 2 B(ax; )

[

a2A

B(a; )

since x was arbitrarily chosen, we have that A

[

a2A

B(a; ), using the compactness

of A; 9 a1; a2:::an 2 A such that A

[n

i=1

B(ai; ) ) A

[n

i=1

B(ai; )

5

since A A.Hence A is totally bounded.

Conversely, suppose A is totally bounded, we show that A is precompact

(i.e A is compact).

Let fangn A, we show that fangn has a Cauchy subsequence

Case I. fan; n 1g is nite

Then there exists n1; ::; np elements of N such that

fan : n 1g = fan1 ; ::; anpg:

Dene for each i 2 f1; ::pg, Ei := fn 2 N : an = anig.

So we have N =

Sp

i=1 Ei.

Since N is innite, then one of the sets Ei is innite. Choose i0 2 f1:; ; :pg such

that Ei0 is innite.

Since Ei0 N, then it has a minimum element. Let m1 := minEi0

For k 1, mk+1 := minfEi0 n fm1;m2:::mkgg

Clearly, mk < mk+1, 8k 2 N, also amk = ani0 since mk 2 Ei0

famkgk1 is a constant subsequence of fang which is convergent.

Case II. fan; n 1g is innite.

For = 1

22 > 0; 9 x1; x2:::xm such that A

m[

i=1

B(xi;

1

22 ) since A is totally

bounded.

This implies that A

m[

i=1

B

(xi;

1

22 ) since B(xi; 1

22 ) B

(xi; 1

22 ).

So A

m[

i=1

B

(xi;

1

22 ) since

m[

i=1

B

(xi;

1

22 ) is closed.

It follows that A

m[

i=1

B(xi;

1

2

) which implies that

fangn

m[

i=1

B(xi;

1

2

), since fangn A

Therefore, 9 i0 2 f1; 2:::mg such that B(xio ; 1

2 ) contains innitely many

terms of the sequence.

I1 := fn 2 N : an 2 B(xio ;

1

2

)g:

I1 is innite and for any fangn2I1 , d(an; am) < 1 8m; n 2 I1.

For 1

23 > 0; 9×1; x2:::xr : A

[r

i=1

B(xi;

1

22 ) so, 9i1 2 f1; 2:::rg such that

B(xi1 ; 1

22 ) contains innitely many terms of the subsequence fangn2I1 .

Now dene I2 :=

n 2 I1 : an 2 B(xi1 ;

1

22 )

:

6

I2 I1 and for n;m 2 I2,

d(an; am) d(an; ai1) + d(am; ai1)

1

22 +

1

22 =

1

2

:

Iteratively, for any Ik innite we can get Ik+1 innite with Ik+1 Ik and

8m; n 2 Ik+1

d(an; am) <

1

2k

Given nk choose nk+1 2 Ik+1 such that nk+1 > nk (this is possible because Ik+1

is innite). Now for j > k the index nj belongs toIk (becauseI1 I2 I3

:::is a nested sequence of sets)

Now for k < j

d(ank ; anj ) d(ank ; ank+1) + : : : + d(anj1 ; anj )

1

2k + 1

2k+1 + : : : + 1

2j1

= 1

2k (1 + 1

2 + : : : + 1

2jk1 )

1

2k1 ! 0; as k ! 1

fankgk1 is a Cauchy subsequence of fang.Its convergence is guaranteed by the

completeness X. Hence A is precompact

Theorem 1.1.15. (Arzela-Ascoli)[6]

A subset A of the space of continuous functions C(K), where K is a non-empty

compact subset of RN, is relatively compact if and only if the two following con-

ditions are satised

i) A is uniformly bounded, i.e there exists M > 0 such that 8x 2 X;

jf(x)j M; 8f 2 K.

ii) A is equicontinuous i.e 8 > 0; 9 > 0:

8 x; y 2 X; kx yk ) jf(x) f(y)j ; 8f 2 K

.

Denition 1.1.16. An inner product (or scalar product) on a vector space X

is a scalar valued function, h ; i on X X such that

i) for each y 2 Xfixed; the functional x 7! hx; yi is linear..

ii) hx; yi = hy; xi; where the bar the complex conjugation.

iii) hx; xi 0; 8 x 2 X.

iv) 8 x 2 X, hx; xi = 0 if and only if x = 0.

7

The pair (X; h ; i) is called Pre-hilbertian (or inner product) space.

The function kkX =

p

hx; xi denes a canonical norm on X.

Denition 1.1.17. A Pre-hilbertian space (H; hi) is called a Hilbert space if

it is complete when equipped with the corresponding canonical norm.

Denition 1.1.18. Cauchy-Schwarz Inequality and parallelogram

law

Let hx; yi be an inner product on a vector space X.

Then

jhx; yij kxkkyk for all x; y 2 X: (1.1.2)

kx + yk2 + kx yk2 = 2(kxk2 + kyk2) for all x; y 2 X (1.1.3)

Proposition 1.1.19. The polarization identity.

Let X be an inner product space. Then for arbitrary x; y 2 X,

hx; yi =

1

4

fjjx + yjj2 jjx yjj2 + ijjx + iyjj2 ijjx iyjj2g (1.1.4)

where i2 = 1.

Theorem 1.1.20. Jordan-Von Neumann.

The norm of a normed linear space X is given by an inner product if and only

if this norm satises the parallelogram law, i.e, if and only if,

kx + yk2 + kx yk2 = 2(kxk2 + kyk2); 8 x; y 2 X:

Denition 1.1.21. Let H be a Hilbert space and x; y 2 H. We say that x is

orthogonal to y denoted by x ? y if hx; yi = 0.

For M H, we say that x is orthogonal to M and write x ? M, if x is

orthogonal to every vector y in M.

The subset of vectors of H ortogonal to M is denoted by

M? = fx 2 H : x ? Mg

and is called the orthogonal complement of M in H.

Proposition 1.1.22. [1] Let M and N be arbitrary subspaces of a Hilbert

space H.Then the following holds

i) M? is a closed subspace of H

ii) M M??,

iii) if M N then N? M?

iv) (M?)? = M.

8

Denition 1.1.23. Given a closed subspace M of H, an operator P dened

on H is called the orthogonal projection onto M if

P(m + n) = m, for all m 2 M and n 2 M?

Theorem 1.1.24. The projection Theorem

Let H be a Hilbert space and M be a closed subspace of H. For an arbitrary

given vector x 2 H, there exists a unique vector m 2 M such that

jjx mjj jjx mjj for allm 2 M:

Furthermore, z 2 M is the unique vector m if and only if

(x z)?M:

Corollary 1.1.25. Direct Sum Decomposition

Let M be a closed subspace of a Hilbert Space, H. Then H = M M?:

Denition 1.1.26. An orthonormal system is a family f’igi2I of elements of

H such that h’i; ‘ji = i j , where, i j is the kronecker delta dened by

i j =

1 i = j;

0 i 6= j:

Example 1.1.27.

ei2nx ; n 2 Z

is an othonormal system for L2([0; 1])

It is easy to see that

hen; emi =

Z

[0;1]

ei2nxei2mxdx

=

Z

[0;1]

ei2(nm)xdx =

1 if n = m

0 if n 6= m:

Denition 1.1.28. [1] A Hilbert space H is separable if H contains a countable

dense subset. Equivalently, a Hilbert space H is said to be separable if there

exists a sequence of vectors v1; v2; :::; vk; ::: which span a dense subspace of H.

Theorem 1.1.29.

A Hilbert space admits a countable orthonormal basis if and only if it is sepa-

rable.

Theorem 1.1.30. Riesz representation theorem

Let f be a continuous linear form on Hilbert space i.e f 2 H. Then there

exist a unique uf 2 H such that hf; vi = hv; uf i for all v 2 H.

Furthermore, we have kfkH = kufkH

9

Denition 1.1.31. Let X and Y be Banach spaces and T 2 B(X; Y ), dene

the dual (also called adjoint) operator as a map T : Y ! X dened by

Tf = f T

that is,

(Tf)(x) = f(T(x)) for all x 2 X:

T is called the (topological) dual or adjoint operator of T.

Remark. T 2 B(Y ;X)

Denition 1.1.32. Adjoint operators on Hilbert spaces

Let T 2 B(H1;H2), the adjoint of T is the unique map T : H2 ! H1 such

that

hTx; yi = hx; Tyi for all x 2 H1 and all y 2 H2

Example 1.1.33. Let be a bounded complex valued Lebesque measurable

function on [a; b]. Let

T : L2([a; b]) ! L2([a; b])

be the bounded linear operator dened by T(f) = f, that is,

(Tf)(t) = (t) f(t) for a:e: t 2 [a; b]:

For all f; g 2 L2([a; b]) we have

hTf; gi =

Z b

a

(Tf)(t)g(t)dt =

Z b

a

(t)f(t)g(t)dt = hf; gi:

Thus T(g) = g :

Theorem 1.1.34. Let T : L2([a; b]) ! L2([a; b]) be the bounded operator

dened by

(Tf)(t) =

Z b

a

G(t; s)f(s)ds

where G is in L2([a; b] [a; b]).

For all g 2 L2([a; b]),

(Tg)(t) =

Z b

a

G(s; t)g(s)ds :

Proof.

10

hTf; gi =

Z b

a

Z b

a

G(t; s)f(s)ds

g(t)dt

=

Z b

a

f(s)

Z b

a

G(t; s)g(t)dt

ds by Fubini’s thorem

= hf; gi

where

g(s) =

Z b

a

G(t; s)g(t)dt

1.2 Complexication of real Banach spaces

Many of the classical Banach functions spaces exist in real or complex-valued

versions. Examples are the Lp()-spaces and C(K)-spaces. Usually one is in-

terested in knowing whether a theory carried for real Banach spaces also holds

for complex Banach spaces (or vice-versa). An approach of solution is given by

the Complexication theory of real Banach spaces. Complexication preserves

norm and allows us to extend all basic notions on any arbitrary real Banach

space to Complex Banach space.

Denition 1.2.1. A complex vector space EC is a complexication of a real

vector space E if the two following conditions holds.

a) There is a one-to-one real linear map j : E ! EC

b) complex-span (j(E)) = EC.

There are, however various alternative concrete descriptions, some of which

include Ordered pair, Tensor and Linear operator descriptions of complexi-

cation.

Hence T is well dened ,infact T 2 B(`2)

Ordered pair description of a complexication.

If E is a real vector space, we can make E E a vector space by dening

(x; y) + (u; v) := (x + u; y + v) 8x; y; u; v 2 E

( + i)(x; y) := (x y; x + y) 8x; y 2 E; 8; 2 R:

11

Consider the map

j : E ! E E

x 7! (x; 0):

Clearly,

j(x + y) = j(x) + j(y) for any x; y 2 E and 2 R;

Ker(j) = fx 2 E : j(x) = (0; 0)g = f0g;

and

Complex span(j(E)) = E E:

The map j satises the conditions a) F(T(B)) ) F(T(B))andb)above; andsothiscomplexvectorspaceisacomplexificationofE:ItisconvenienttodenoteitbyEEiE:andalsosuppressrefrencetojbywrittingz = x+iy for the element z =

(x; y) = j(x) + ij(y):Itisnaturaltowritex=<e z and y = =mz.

For other two descriptions, we refer to[4].

Denition 1.2.2. Let E be a real Banach space and EC := EiE.

EC; jj jj

is called a complexication of E if

EC; jjjj

is a complex Banach space, jjjjjE

is the original norm of E (i.e jjx + i0jj = jjxjj; 8x 2 E) and

jjx + iyjj = jjx iyjj 8x; y 2 E:

Now we might ask the Question: Is there a norm on EC which makes EC

a complex Banach space and induces the original norm onE ?

The answer is armative and there are innitely many ways to do so.[4].

Proposition 1.2.3. Let EC be a complexication of the real space E endowed

with a norm jj jj such that (E; jj jj) is Banach. Then jj jjT as dened below

denes a norm on EC.

jjx + iyjjT := sup

0t2

jj(cos t)x (sin t)yjj

All other complexication norms jj jj on EC are equivalent to jj jjT . Indeed

jjx + iyjjT jjx + iyjj 2jjx + iyjjT 8 x; y 2 E: (1.2.1)

Denition 1.2.4. Let E be a real Banach space. We say that a norm on the

complexication EC is reasonable if

c) jjj(x)jj = jjxjj 8x 2 E

d) jjx + iyjj = jjx iyjj x; y 2 E

When EC is equipped with such a norm, we call it a reasonable complexication

of E

12

Proposition 1.2.5. Let EC be a reasonable complexication of the real Banach

space E. for any x; y 2 E we have jjxjjE jjx+iyjjEC and jjyjjE jjx+iyjjEC

Proof. By property (c),

2jjxjjE = jj(x + iy) + (x iy)jjEC jjx + iyjjEC + jjx iyjjEC

An application of property d) gives jjxjjE jjx + iyjjEC

Similarly we have the other inequality.

Proposition 1.2.6. Let EC be a complexication of the real Banach space E

For any x; y 2 E we have,

sup

0t2

jj(cos t)x (sin t)yjjE jjx + iyjjEC

and

jjx + iyjjEC inf

0t2

(jj(cos t)x (sin t)yjjE + jj(sin t)x + (cos t)yjjE):

Proof. For each 0 t 2

jjx+iyjjEC = jjeit(x+iy)jjEC = jj((cos t)x(sin t)y)+i((sin t)x+(cos t)y)jjEC.

Using proposition (1.2.5) on the left and the triangle inequality on the right,

we have

jj(cos t)x (sin t)yjjE jjx + iyjjEC and jjx + iyjjEC jj(cos t)x (sin t)yjjE +

jj(sin t)x + (cos t)yjjE.

Hence, the result follows immediately.

Let us check verify that jj jjT is a reasonable complexication norm.

i)For any x 2 E, jjxjjT = jjx + i0jjT = sup

0t2

jjxcost 0sintjj = jjxjj

ii) jjxcost ysintjj = jjxcos(t) + ysin(t)jj and the function

t 7! xcost ysint is periodic with period of 2 for all x; y 2 E.Therefore,

jjx + iyjjT = sup

0t2

jjxcost ysintjj

= sup

t2R

jjxcos(t) + ysin(t)jj

= sup

t2R

jjxcos(t) + ysin(t)jj

= sup

0t2

jjxcost + ysintjj

jjx + iyjjT = jjx iyjjT

We have shown that property c) and d) are satised. Hence jjjjT is a reasonable

complexication norm.

Let us also verify the inequality in (1.2.1)

From proposition (1.2.6)

13

jjx + iyjjT jjx + iyjj inf

0t2

(jjxcost ysintjjE + jjxsint + ycostjjE)

sup

0t2

jjxcost ysintjjE + sup

0t2

jjxsint + ycostjjE

= 2jjx + iyjjT

jjx + iyjjT jjx + iyjj 2jjx + iyjjT

The norm jj jjT was rst considered by A.Y Taylor [ad]. (EC; jj jjT ) is known

as Taylor complexication of E.

There is a useful alternative description of jjx + iyjjT :

jjx + iyjjT = sup

0t2

jjxcost ysintjj

= sup

0t2

sup

jjfjjE1

jf(x)cost f(y)sintj

= sup

jjfjjE1

p

f(x)2 + f(y)2 ; 8x; y 2 E

Another feature of the Taylor complexication, is that it is a general complex-

ication whose denition is not tied to any specic characteristic of the real

Banach space E which is being complexied. Moreover, this procedure allows

us to extend continuous linear maps between real Banach space to complex

linear maps between their complexications without increasing the norm. If

L : E ! F is a linear map between real vector spaces E and F, there is a

unique complex-linear extension ~L : EC ! FC given by

~L

(x + iy) = L(x) + iL(y)

Proposition 1.2.7. Let E and F be real Banach spaces. If L 2 L(E; F), then

~L

2 L

(EC; jj jjT ); (FC; jj jjT )

and jj~ Ljj = jjLjj

Proof. Since ~L extends L, we have jj~Ljj jjLjj.

On the other hand, if x; y 2 E then

jj~L(x + iy)jjT = jjL(x) + iL(y)jjT = sup

0t2

jjL(x)cost L(y)sintjjF

= sup

0t2

jjL(xcost ysint)jjF

jjLjj sup

0t2

jjxcost ysintjjE

jj~L(x + iy)jjT jj~ Ljjjjx + iyjjT =) jj~Ljj jjLjj

Hence,

jj~Ljj = jjLjj

14

Taylor’s procedure is just one of innitely many procedures with similar

properties.

1.3 Some function spaces (Lp, Sobolev spaces)

Denition 1.3.1. Let 1 p < 1;

be an open bounded subset of Rn. We

dene

Lp(

) as the set of measurable functions f :

! R such that

Z

jf(x)jpdx < +1

L1(

) as the set of measurable functions f :

! R such that esupjfj < 1

where,

esupjfj = inffk > 0; jf(x)j k a:e x 2

g

For f 2 Lp(

), we dene,

jjfjjp =

Z

jf(x)jp

1

p ; 1 p < 1:

jjfjj1 = esupjfj; if p = 1:

Theorem 1.3.2. The following properties holds for Lp space

i) Lp-space is Banach for 1 p 1

ii) Lp-space is Re exive for 1 < p < 1

iii) Lp-space is Separable for 1 p < 1

F(T(B)) ) F(T(B))Itisalsoexpedienttorecallsomenotationsandbasicresultsfromdistributiontheory:

Denition 1.3.3. The Space L10

(

) is the space of all Lebesgue measurable

functions in

having absolute value integrable on each compact subset of

A multi-index is a vector (1; 2:::n) 2 Nn.

The length of is given by jj = 1 + ::: + n

We also dene the generalized derivative

D =

@jj

@1×1 : : :@nxn

Denition 1.3.4.

A locally integrable function v i.e element of L10

(

) is called the th weak

derivative of u 2 L10

(

), if it satises

Z

u(x)D(x)dx = (1)jj

Z

v(x)(x)dx; 8 2 D(

)

Where D(

) denotes the set of C1-functions on

with compact support in

15

Let x 2 Rn, we write x = (x0; xn) with x0 2 Rn1, x0 = (x1; x2; :::xn1). We

consider the following notations[5].

Rn

+ = fx = (x0; xn) 2 Rn : xn > 0g

B = fx = (x0; xn) 2 Rn : jjx0jj < 1; jxnj < 1g

B+ = fx = (x0; xn) 2 B : xn > 0g

B0 = fx = (x0; xn) 2 B : xn = 0g

Denition 1.3.5. We say that an open subset

Rn is of class Cm(

)(m; integer)

if for every x 2 @

, there exist an open neighbourhood U of x in Rn and a

map : B ! U such that,

i) is a bijection

ii) 2 Cm( B; U);1 2 Cm(U

; B)

iii) (B+) =

\ U; (B0) = @

\ U

Denition 1.3.6.

Let 1 p +1, m 2 N,. The Sobolev space Wm;p(

) is dened by

Wm;p(

) = fu 2 Lp(

) j Du 2 LP (

) for all jj mg

We shall be working with the case p = 2 .The Sobolev space Wm;p(

) are

denoted by Hm(

). H1

0(

) is the closure of D(

) in H1(

)

. Finally, we shall consider two important results which are very instrumental

to the application of spectral theorem of compact self adjoint operators to elliptic

partial dierential equations.

Proposition 1.3.7. Poincare Inequality

Let 1 p < 1 and

a bounded open subset of RN. Then there exist a

constant C(

; p) such that

jjujjLp(

) CjjrujjLp(

) 8u 2 W1;p

0 (

):

If

is connected and satises a C1 boundary condition, then there exists a

constant C(

; p) such that

jju ujjLp(

) CjjrujjLp(

); 8u 2 W1;p(

)

u =

1

j

j

Z

u(x)dx

, is the mean value of u on

16

Denition 1.3.8.

Let E and F be two normed vector spaces such that E F.We say Ec ,! F is

compact embeddings if any bounded subset of E is precompact in F ,or equiv-

alently any bounded sequence of E has a subsequence that converges in F.

17

CHAPTER