seminorm

seminorm

Let

V

be a real, or a

complex vector space

, with

K

denoting the
corresponding field of scalars. A

seminorm

is a

function

p

:

V

→

ℝ

+

,

from

V

to the set of non-negative

real numbers

, that

satisfies

the
following two properties.

p

⁡

(

k

⁢

𝐮

)

=

|

k

|

⁢

p

⁡

(

𝐮

)

,

k

∈

K

,

𝐮

∈

V

Homogeneity

p

⁡

(

𝐮

+

𝐯

)

≤

p

⁡

(

𝐮

)

+

p

⁡

(

𝐯

)

,

𝐮

,

𝐯

∈

V

,

Sublinearity

A seminorm differs from a norm in that it is permitted that

p

⁡

(

𝐮

)

=

0

for some non-zero

𝐮

∈

V

.

It is possible to characterize the seminorms properties
geometrically. For

k

>

0

, let

B

k

=

{

𝐮

∈

V

:

p

⁡

(

𝐮

)

≤

k

}

denote
the ball of radius

k

. The homogeneity property is

equivalent

to the
assertion that

B

k

=

k

⁢

B

1

,

in the sense that

𝐮

∈

B

1

if and only if

k

⁢

𝐮

∈

B

k

.

Thus, we see that a seminorm is fully determined by its unit ball.
Indeed, given

B

⊂

V

we may define a function

p

B

:

V

→

ℝ

+

by

p

B

⁡

(

𝐮

)

=

inf

⁡

{

λ

∈

ℝ

+

:

λ

-

1

⁢

𝐮

∈

B

}

.

The geometric nature of the unit ball is described by the following.

Proposition 1

The function

p

B

satisfies the homegeneity property if and only if
for every

u

∈

V

, there exists a

k

∈

R

+

∪

{

∞

}

such that

λ

⁢

𝐮

∈

B

if and only if

∥

λ

∥

≤

k

.

Proposition 2

Suppose that

p

is

homogeneous

. Then, it is sublinear if and
only if its unit ball,

B

1

, is a convex subset of

V

.

Proof.

First, let us suppose that the seminorm is both sublinear
and homogeneous, and prove that

B

1

is necessarily convex. Let

𝐮

,

𝐯

∈

B

1

, and let

k

be a real number between

0

and

1

.
We must show that the

weighted average

k

⁢

𝐮

+

(

1

-

k

)

⁢

𝐯

is in

B

1

as
well. By

assumption

,

p

⁡

(

k

⁢

𝐮

+

(

1

-

k

)

⁢

𝐯

)

≤

k

⁢

p

⁡

(

𝐮

)

+

(

1

-

k

)

⁢

p

⁡

(

𝐯

)

.

The right side is a
weighted average of two numbers between

0

and

1

, and is therefore
between

0

and

1

itself. Therefore

k

⁢

𝐮

+

(

1

-

k

)

⁢

𝐯

∈

B

1

,

as desired.

Conversely

, suppose that the seminorm function is homogeneous, and
that the unit ball is convex. Let

𝐮

,

𝐯

∈

V

be given, and let us show that

p

⁡

(

𝐮

+

𝐯

)

≤

p

⁡

(

𝐮

)

+

p

⁡

(

𝐯

)

.

The essential complication here is that we do not exclude the
possibility that

p

⁡

(

𝐮

)

=

0

, but that

𝐮

≠

0

.
First, let us consider the case where

p

⁡

(

𝐮

)

=

p

⁡

(

𝐯

)

=

0

.

By homogeneity, for every

k

>

0

we have

k

⁢

𝐮

,

k

⁢

𝐯

∈

B

1

,

and hence

k

2

⁢

𝐮

+

k

2

⁢

𝐯

∈

B

1

,

as well. By homogeneity, again,

p

⁡

(

𝐮

+

𝐯

)

≤

2

k

.

Since the above is true for all positive

k

, we infer that

p

⁡

(

𝐮

+

𝐯

)

=

0

,

as desired.

Next suppose that

p

⁡

(

𝐮

)

=

0

, but that

p

⁡

(

𝐯

)

≠

0

. We will show
that in this case, necessarily,

p

⁡

(

𝐮

+

𝐯

)

=

p

⁡

(

𝐯

)

.

Owing to the homogeneity assumption, we may without loss of generality
assume that

p

⁡

(

𝐯

)

=

1

.

For every

k

such that

0

≤

k

<

1

we have

k

⁢

𝐮

+

k

⁢

𝐯

=

(

1

-

k

)

⁢

k

⁢

𝐮

1

-

k

+

k

⁢

𝐯

.

The right-side expression is an

element of

B

1

because

k

⁢

𝐮

1

-

k

,

𝐯

∈

B

1

.

Hence

k

⁢

p

⁡

(

𝐮

+

𝐯

)

≤

1

,

and since this holds for

k

arbitrarily close to

1

we conclude that

p

⁡

(

𝐮

+

𝐯

)

≤

p

⁡

(

𝐯

)

.

The same

argument

also shows that

p

⁡

(

𝐯

)

=

p

⁡

(

-

𝐮

+

(

𝐮

+

𝐯

)

)

≤

p

⁡

(

𝐮

+

𝐯

)

,

and hence

p

⁡

(

𝐮

+

𝐯

)

=

p

⁡

(

𝐯

)

,

as desired.

Finally, suppose that neither

p

⁡

(

𝐮

)

nor

p

⁡

(

𝐯

)

is zero. Hence,

𝐮

p

⁡

(

u

)

,

𝐯

p

⁡

(

v

)

are both in

B

1

, and hence

p

⁡

(

u

)

p

⁡

(

u

)

+

p

⁡

(

v

)

⁢

𝐮

p

⁡

(

u

)

+

p

⁡

(

v

)

p

⁡

(

u

)

+

p

⁡

(

v

)

⁢

𝐯

p

⁡

(

v

)

=

𝐮

+

𝐯

p

⁡

(

u

)

+

p

⁡

(

v

)

is in

B

1

also. Using homogeneity, we conclude that

p

⁡

(

u

+

v

)

≤

p

⁡

(

u

)

+

p

⁡

(

v

)

,

as desired.

Title

seminorm

Canonical name

Seminorm

Date of creation

2013-03-22 12:24:57

Last modified on

2013-03-22 12:24:57

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

20

Author

rmilson (146)

Entry type

Definition

Classification

msc 46B20

Synonym

semi-norm

Defines

homogeneous

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