Normed Linear Spaces notes by Dr E_Prempeh_Ghana

8/13/2019 Normed Linear Spaces notes by Dr E_Prempeh_Ghana

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NORMED LINEAR SPACES

1.1 Normed Linear Spaces

Definition 1.1 (Linear or Vector Space)

Let .X ,K a scalar field (usually ,K the real line). Suppose the two functions

, ,

:X X X

:K X X

called respectively addition and scalar multiplication are defined, (that is, for any

, ,x y X scalar ,K x y X and x X )

such that

1. X is an abelian group( that is, for any , , ,x y z X

x y y x

( ) ( )x y z x y z

0 X : 0x x

,x X ( )x X : ( ) 0x x )

2. ( )x y x y , , ,x y X K 3. ( ) x x x , , ,K x X 4. ( ) ( )x x , , K , x X 5. 1 x x for 1 K , x X

Then X is called a linear space or a vector space over .K If K is the set of real

numbers, X is called a real linear space or if K is the set of complex numbers, X is

called a complex linear space.

(Notation: We take x x ).


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Definition 1.2 (Normed Linear Space)

Let X be a linear space over K.

A norm on X is a real-valued function , where

: 0,X such that for any , ,x y X K the following conditions are satistied:

1:N 0x and 0x if and only if 0x

2 :N x x , K , x X

3:N x y x y , ,x y X

A linear space with a norm defined on it is called a Normed linear space.

Examples of Normed linear spaces

Example 1.3

Let2

X (the plane). For any 2. ,x y X 1 2( : ( , ),x x x 1 2( , )),y y y scalar

,

Define1 1 2 2

: ( , )x y x y x y

1 2: ( , )x x x

With these definitions, (easy, exercise!), 2 is a Vector Space!

For each2

1 2( , ) ,x x x define

2: 0,

by

1 21 2

max , maxi

ix x x x

W.T.S:2

( , )

is a normed linear space.


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Verification

1:N Absolute value of any real number ,ix 0,ix

Thus

1 2max 0i

ix

i.e.1 1

: max 0i

x

x x

Next, suppose 1 2, 0x x x

W.T.S: 0x

Now1 2

( , ) 0x x x 1 0x , 2 0x (in Cartesian Coordinate System)

and so1 2max 0i

ix

Next, suppose 0.x

W.T.S : 0x

Now 0x

means1 2max 0i

ix

Without loss of generality (wilog),

let 11 2

maxi

i

x x

.

Then 1 0x implies 2 0x

(since 1 20 x x and none is negative).

Hence 1 2 0x x 1 2 0x x

So,1 2( , ) (0,0) 0.x x x

2 :N 1 2 1 2( , ) ( , )x x x x x

1 2 1 21 2

: max max maxi i ii i

i

x x x x

i.e. .x x


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3:N Let 1 2( , ),x x x 1 2( , ),y y y

Then 1 2 1 2( , ) ( , )x y x x y y

1 1 2 2( , )x y x y

1 1 2 21 2

max , maxi i

ix y x y x y

1 2max( )i i

ix y

1 2 1 2max max

i ii i

x y x y

2( , ) ( , )x

is a normed linear space.

Example 1.4. Let2.X For any 1 2( , )x x x X ,

Define 2

: 0,X by1

2 2 21 22

( )x x x

W.T.S:2

2( , ) is a normed linear space.

Verification

1N : Trivial (Exercise!!!)2 :N 1 2 1 22 2( , ) ( , )x x x x x

1 1

2 2 2 2 2 22 21 2 1 2

: ( ) ( ) ( )x x x x

1 12 2 2 2 22 2

1 2 1 2( ) ( )x x x x

2x

3:N (Recall Cauchy Schwartz Inequality for n )

For 1

n

i ix

,

1

n

i iy

,

n

1 12 2

2 2

1 1 1

.n n n

i i i i

i i i

x y x y


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This is used to verify 3N as follows:

Let1 2

( , ),x x x 1 2( , )y y y 2

Then

22 2 2

1 1 2 22 21

( , ) : ( )i i

i

x y x y x y x y

22 2

1

( 2 )i i i i

i

x x y y

2 2 22 2

1 1 12i i i ii i ix x y y

12 2 2 22

2 2 2 2

1 1 1 1

2i i i i

i i i i

x x y y

(Cauchy Schwartz)

2 22A AB B

(where

12 2

2

21

i

i

A x x

,

12 2

2

21

i

i

B y y

2 2

2 2( ) ( )A B x y

2 2

2 2 2( )x y x y

(If ( 2

1( (# ))ve 22( (# ))ve 1 2(# ) (# )ve ve )

2 2 2

x y x y

2

2( , ) is a normed linear space.

Example 1.5

Let2

,X for any 21 2( , ) ,x x x

define2

11

:i

i

x x


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W.T.S :2

1( , ) is a normed linear space. (Exercise)

The Spacesn

p (1 p ) andn

Letn

X and define the map

: 0,np

X

such that for any 1 2, ,..., n

nx x x x

1

1 2

1

: , ,..., : ,n p

p

n ip pi

x x x x x

1 p

(In example 1.5, 1,p 2n ), and if p ,

1: max

ii n

x x

The verification that

is a norm is easy. (Exercise!)

So we verify the case of the mapp

1:N For any 1 2( , ,..., ) n

nx x x x

1

1

:n p

p

ipi

x x

, (1 ),p ix for 1 i n

Clearly, 0,p

ix

1

0n

p

i

i

x

and1

1

0,n p

p

i

i

x

implying 0.px

Assume 0p

x

1

1

0n p

p

i

i

x

1

0n

p

i

i

x

0p

ix

0,ix 1 i n 0x ,

Next, assume 0x 0ix 1 i n 1

0n

p

i

i

x

0px


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2 :N For any ,

1 1 1

1 2

1 1 1

, ,..., :n n np p p

p p pp

n i i ip pp i i i

x x x x x x x x

3:N We need Hlders Inequality: For , ,n

x y 1 2( , ,..., )nx x x x and

1 2( , ,..., ),ny y y y

1 1

1 1 1

n n np qp q

i i i i

i i i

x y x y

where1 1

1p q

, and thus ( , )p q is called the conjugate pair. If 2,p q

the Hlders Inequality becomes Cauchy-Schwartz Inequality. This Hlders inequality is

used to verify 3N as follows: for , ,nx y

1 1 2 21

, ,...,n

p pp

n n i ip pi

x y x y x y x y x y

1

1

np

i i i i

i

x y x y

1 1

1 1

n np p

i i i i i i

i i

x x y y x y

Using the Hlders inequality on each of the terms on the r. h. s, we have

1 1 1 1

1 1

1 1 1 1

n n n np q p qp p q p p qp

i i i i i ipi i i i

x y x x y y x y

1

1

1

( )n q

p q

i ip pi

x y x y

From1 1

1,p q

1 1p

p q

1p q p


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1

1

n qpp

i ip p pi

x y x y x y

1

1

pn p qp

i ip pi

x y x y

p

q

p p px y x y

1

1pq

p p px y x y

p p p

x y x y (since1 1

1q p

)

Remark 1.7

The spacen

with norm,

is denoted byn

, while the space

n with norm

p

is denoted byn

p . The notations n

p andn

p are used to emphasize the type of

n tuple as follows:

:np

1

1 2

1

, ,..., : , :np

n pp

n i i

i

x x x x x x x

1

1 2

1

: , ,..., : , :np

n ppn

p n i i

i

z z z z z z z

The spacen

with norm2

is called the Euclidean Space and the spacen

with

norm2

is called the Unitary Space.


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Next we consider the space p of infinite sequences 1i ix

such that

1

,

p

i

i

x

1 p , that is

1 21

: , ,... , : p

p i i

i

x x x x x

We now study the spacep into some detail.

For 1p ,

1 1 2 31

, , ,... :i

i

x x x x x

If 1, 1,0,0,0,...y is 1 ?y

We notice,1

1 1 0 0 0 ... 2 .ii

y

So 1y .

If1 1 1

1, , , ,...2 3 4

w

, is

1?w

Again, we compute1

1 1 1

1 ...2 3 4i

iw

. This series is called the Harmonic

Series which certainly does not converge, that is,1

i

i

w

is false. So 1.w l

For 2,p

2

2 1 2 3

1

: , , ,... : ,i ii

x x x x x x

For ,y w, considered above, 2y and 2.w So 1 2.

One can generalize to have:

Proposition 1.8

If p q , then p q .

Proof : (Proof by contradiction)


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Assuming by contradiction let p q , and let 1, 2p q , and2 1

but by the above illustration,1 2 ; a contradiction! Therefore for ,p q p q .

Exercise 1.9: (The Space, p )

The space

p 1 21

(1 ) , ,... : p

i

i

p x x x x

with addition and scalar multiplication

defined in an obvious way, is a normed linear space with norm defined by

1

1

:p

pp

i

i

x x

for 1 2 3, , ,... .px x x x

Exercise: Verify that , ,p p 1 p is a normed linear space.

The space is defined by

: { 1 2 3, , ,... , :ix x x x x x is bounded},

That is, is the space of all bounded sequences. This means that if

1 2 3, , ,...x x x x then there exists a real constant M such that ix M i .

Consider 1,1, 1,1, 1,1,... ,y then y . Observe that 1y , 2y for

1 1 11, , , ,... ,

2 3 4

.

Example 1.10 (The Space )

Let :X { 1 2 3, , ,... : ,ix x x x x M for some real constant M }

Define

: 0,


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by

1

supi

i

x x

Exercise: Verify that , ll is a normed linear space.

Next, consider the sequence space c defined by

:c { 1 2 1, ,... : i ix x x x

converges}

Since a convergent sequence of numbers is necessarily bounded, then c is a subspace

of . To see that c is a proper subspace of ,

it is observed that the vector

1,1, 1,1,...y (since y is bounded)

but

y c (since y is not convergent).

It is therefore proper to define the same norm on c as defined for .

Example 1.11 (The Space, c )

Let :c { 1 21

, ,... :i

i

x x x x

converges}. For any 1 2, ,...x x x c ,

define

1

: supic

i

x x

Then , cc is a normed linear space (Exercise!!)

Example 1.12 (The Space, 0c )

Let 0 1 2: ( , ,...) : 0 .n

nc x x x x For any 0x c , define

0 1

: supic

i

x x

then 0

0,

cc is a normed linear space. Clearly 0c is a proper subspace c .


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Example 1.13

Let , ,X C a b the space of all real-valued continuous functions: , ,f g which are

functions of an independent variable , ,t a b closed and bounded interval. For any

, , ,f g C a b define

: ,f g t f t g t , ,f g C a b

and ,f t f t ,K ,f C a b

With these definitions ,C a b is a vector space. (Verify)

For arbitrary ,f C a b , define

,: sup ( )

t a b

f f t

1: ( )

b

af f t dt

1

22

2: ( )

b

af f t dt

Then, ,X , 1, ,X and 2, ,X are normed linear spaces. The proofs are

left as exercises.

Exercise 1.1

1. Let ,X C a b be the space of all continuous real-valued functions on ,a b whichalso have continuous first order derivatives on , .a b For any , ,f C a b define

: max maxa t b a t b

df tf f t

dt

Verify that ,X with obvious addition and scalar multiplication, is a normed linear

space.


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2. Let ,X C a b be the space of all real-valued continuous functions on , .a b Forany ,f X define

: .b

a

f f t dt

Verify that X with addition and scalar multiplication as defined in Example 1.13 is

a normed linear space.

3. Let ,X C a b be the space of all continuous real-valued functions on ,a b with

122

:b

af f t dt

for any , .f C a b Verify that X is a normed linear space.

(Hint: Use Cauchy-Schwartz Inequality for Integration: For any , ,f g C a b )

1 1

2 22 2b b b

a a afg dt f t dt g t dt

4. Let X be a normed linear space. Prove that for any ,x y X a. x y x y b. The mapping x x is continuous ( in the sense that if

nx x , then

nx x )

c. Addition and scalar multiplication are jointly continuous, that is, ifn

x x and

ny y , then n nx y x y and if nx x and na a , then n na x ax

as n , where na and a are real numbers.

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Normed Linear Spaces notes by Dr E_Prempeh_Ghana