equivalent norms

equivalent norms

Let

∥

x

∥

and

∥

x

∥

′

be two norms on
a

vector space

V

. These norms are

equivalent norms

if
there exists a number

C

>

1

such that

1

C

⁢

∥

x

∥

≤

∥

x

∥

′

≤

C

⁢

∥

x

∥

(1)

for all

x

∈

V

.

Since equation (

1

) is

equivalent

to

1

C

⁢

∥

x

∥

′

≤

∥

x

∥

≤

C

⁢

∥

x

∥

′

(2)

it follows that the definition is well defined. In other words,

∥

⋅

∥

and

∥

⋅

∥

′

are equivalent if and only if

∥

⋅

∥

′

and

∥

⋅

∥

are equivalent.
An alternative condition is that there exist positive real
numbers

c

,

d

such that

c

⁢

∥

x

∥

≤

∥

x

∥

′

≤

d

⁢

∥

x

∥

.

However, this condition is equivalent to the above
by setting

C

=

max

⁡

{

1

/

c

,

d

}

.

Some key results are as follows:

1.

If

γ

>

0

and

∥

x

∥

′

=

γ

⁢

∥

x

∥

, then

∥

⋅

∥

and

∥

⋅

∥

′

are equivalent. For example,
if

γ

>

1

, then condition (

1

) holds with

C

=

γ

, and
for

γ

<

1

, condition (

2

) holds with

C

=

1

/

γ

.

2.

Suppose norms

∥

⋅

∥

and

∥

⋅

∥

′

are equivalent norms
as in equation (

1

), and let

B

r

⁢

(

x

)

and

B

r

′

⁢

(

x

)

be the
open balls with respect to

∥

⋅

∥

and

∥

⋅

∥

′

, respectively.
By this result (

http://planetmath.org/ScalingOfTheOpenBallInANormedVectorSpace

)
it follows that

C

⁢

B

ε

⁢

(

x

)

⊆

B

ε

′

⁢

(

x

)

⊆

1

C

⁢

B

ε

⁢

(

x

)

.

It follows that the

identity map

from

(

V

,

∥

⋅

∥

)

to

(

V

,

∥

⋅

∥

′

)

is a

homeomorphism

. Or, alternatively, equivalent norms on

V

induce the same
topology on

V

.

3.

The

converse

of the last paragraph is also true, i.e. if two norms induce the same topology on

V

then they are equivalent. This follows from the fact that every

continuous

linear function

between two

normed vector spaces

is

bounded

(

http://planetmath.org/BoundedOperator

) (see this entry (

http://planetmath.org/BoundedOperator

)).

4.

Suppose

⟨

⋅

,

⋅

⟩

and

⟨

⋅

,

⋅

⟩

′

are

inner product

. Suppose further that the

induced norms

∥

⋅

∥

and

∥

⋅

∥

′

are equivalent as in equation

1

. Then, by the

polarization identity

, the inner products

satisfy

1

C

2

⁢

⟨

v

,

w

⟩

′

≤

⟨

v

,

w

⟩

≤

C

2

⁢

⟨

v

,

w

⟩

.

5.

On a finite dimensional vector space all norms are equivalent
(see this page (

http://planetmath.org/ProofThatAllNormsOnFiniteVectorSpaceAreEquivalent

)).
This is easy to understand as the

unit sphere

is

compact

if and only if
a space is finite dimensional.
On infinite dimensional spaces this result does not hold (see
this page (

http://planetmath.org/AllNormsAreNotEquivalent

)).

It follows that on a finite dimensional vector space,
one can check

continuity

and convergence with respect with any norm.
If a

sequence

converges

in one norm, it converges in all norms.
In matrix analysis this is particularly useful as one can choose the norm that
is most easily calculated.

6.

The

concept

of equivalent norms also generalize to possibly non-symmetric norms. In this setting, all norms are also equivalent on a finite dimensional vector space. In particular,

∥

⋅

∥

and

∥

-

⋅

∥

are
equivalent, and there exists

C

>

0

such that

∥

-

v

∥

≤

C

⁢

∥

v

∥

,

v

∈

V

.

Title

equivalent norms

Canonical name

EquivalentNorms

Date of creation

2013-03-22 13:39:28

Last modified on

2013-03-22 13:39:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Definition

Classification

msc 46B99

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