extended norm

extended norm

It is sometimes convenient to allow the norm to take

extended real numbers

as values. This way, one can accomdate elements of

infinite

norm in one’s

vector space

. The formal definition is the same except that one must take care in stating the second condition to avoid the

indeterminate form

0

⋅

∞

.

Definition:

Given a real or

complex vector space

V

, an

extended norm

is a map

∥

⋅

∥

→

ℝ

¯

which staisfies the follwing three defining properties:

1.

Positive definiteness:

∥

v

∥

>

0

unless

v

=

0

, in which case

∥

v

∥

=

0

.

2.

Homogeneity:

∥

λ

⁢

v

∥

=

|

λ

|

⁢

∥

v

∥

for all non-zero scalars

λ

and all vectors

v

.

3.

Triangle inequality

:

∥

u

+

v

∥

≤

∥

u

∥

+

∥

v

∥

for all

u

,

v

∈

V

.

Example

Let

C

0

be the space of

continuous functions

on the real line. Then the

function

∥

⋅

∥

defined as

∥

f

∥

=

sup

x

⁡

|

f

⁢

(

x

)

|

is an extended norm. (We define the

supremum

of an

unbounded set

as

∞

.) The reason it is not a norm in the strict sense is that there exist continuous functions which are unbounded. Let us check that it

satisfies

the defining properties:

1.

Because of the

absolute value

in the definition, it is obvious that

∥

f

∥

≥

0

for all

f

. Furthermore, if

∥

f

∥

=

0

,

then

sup

x

⁡

|

f

⁢

(

x

)

|

=

0

, so

|

f

⁢

(

x

)

|

≤

0

for all

x

, which implies that

f

=

0

.

2.

If

f

is

bounded

, then

∥

f

∥

=

sup

x

|

λ

f

(

x

)

|

=

sup

x

|

λ

|

|

f

(

x

)

|

=

|

λ

|

sup

x

|

f

(

x

)

|

=

|

λ

|

∥

∥

f

∥

If

f

is unbounded and

λ

≠

0

, then

λ

⁢

|

f

|

is unbounded as well. By the convention that

infinity

times a finite number is infinity, property (2) holds in this case as well.

3.

If both

f

and

g

are bounded, then

∥

f

+

g

∥

=

sup

x

⁡

|

f

⁢

(

x

)

+

g

⁢

(

x

)

|

≤

sup

x

⁡

(

|

f

⁢

(

x

)

|

+

|

g

⁢

(

x

)

|

)

≤

sup

x

⁡

|

f

⁢

(

x

)

|

+

sup

x

⁡

|

g

⁢

(

x

)

|

=

∥

f

∥

+

∥

g

∥

On the other hand, if either

f

or

g

is unbounded, then the right hand side of (3) is infinity by the convention that anything other minus infinity added to plus infinity is still infinity and infinity is bigger than anything which could appear on the left.

An extended norm

partitions

a vector space into

equivalence classes

modulo the

relation

∥

u

-

v

∥

<

∞

. The equivalence class of zero is a vector space consisting of all elemets of finite norm. Restricted to this equivalence class the extended norm reduces to an ordinary norm. The other equivalence classes are

translates

of the equivalence class of zero. Furthermore, when we use the distance function of the norm to define a

topology

on our vector space, these equivalence classes are exactly the

connected components

of the topology.

Title

extended norm

Canonical name

ExtendedNorm

Date of creation

2013-03-22 14:59:49

Last modified on

2013-03-22 14:59:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 46B99

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