ring of continuous functions

ring of continuous functions

Let

X

be a

topological space

and

C

⁢

(

X

)

be the

function space

consisting of all

continuous functions

from

X

into

ℝ

, the reals (with the usual

metric topology

).

Ring Structure on

C

⁢

(

X

)

To formally define

C

⁢

(

X

)

as a ring, we take a step backward, and look at

ℝ

X

, the set of all

functions

from

X

to

ℝ

. We will define a ring

structure

on

ℝ

X

so that

C

⁢

(

X

)

inherits that structure and forms a ring itself.

For any

f

,

g

∈

ℝ

X

and any

r

∈

ℝ

, we define the following

operations

:

1.

(

addition

)

(

f

+

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

+

g

⁢

(

x

)

,

2.

(multiplication)

(

f

⁢

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

,

3.

(

identities

) Define

r

⁢

(

x

)

:=

r

for all

x

∈

X

. These are the

constant functions

. The special constant functions

1

⁢

(

x

)

and

0

⁢

(

x

)

are the

multiplicative

and

additive identities

in

ℝ

X

.

4.

(

additive inverse

)

(

-

f

)

⁢

(

x

)

:=

-

(

f

⁢

(

x

)

)

,

5.

(

multiplicative inverse

) if

f

⁢

(

x

)

≠

0

for all

x

∈

X

, then we may define the multiplicative inverse of

f

, written

f

-

1

by

f

-

1

⁢

(

x

)

:=

1

f

⁢

(

x

)

.

This is not to be confused with the

functional

inverse

of

f

.

All the ring axioms are easily verified. So

ℝ

X

is a ring, and actually a

commutative ring

. It is immediate that any constant function other than the additive identity is

invertible

.

Since

C

⁢

(

X

)

is closed under all of the above operations, and that

0

,

1

∈

C

⁢

(

X

)

,

C

⁢

(

X

)

is a

subring

of

ℝ

X

, and is called

the

ring of continuous functions

over

X

.

Additional Structures on

C

⁢

(

X

)

ℝ

X

becomes an

ℝ

-

algebra

if we define scalar multiplication by

(

r

⁢

f

)

⁢

(

x

)

:=

r

⁢

(

f

⁢

(

x

)

)

. As a result,

C

⁢

(

X

)

is a

subalgebra

of

ℝ

X

.

In addition to having a ring structure,

ℝ

X

also has a natural order structure, with the partial order defined by

f

≤

g

iff

f

⁢

(

x

)

≤

g

⁢

(

x

)

for all

x

∈

X

. The

positive cone

is the set

{

f

∣

0

≤

f

}

. The

absolute value

, given by

|

f

|

⁢

(

x

)

:=

|

f

⁢

(

x

)

|

, is an

operator

mapping

ℝ

X

onto its positive cone. With the absolute value operator defined, we can put a

lattice

structure (

http://planetmath.org/Lattice

) on

ℝ

X

as well:

•

(meet)

f

∨

g

:=

2

-

1

⁢

(

f

+

g

+

|

f

-

g

|

)

. Here,

2

-

1

is the constant function valued at

1

2

(also as the multiplicative inverse of the constant function

2

).

•

(join)

f

∧

g

:=

f

+

g

-

(

f

∨

g

)

.

Since taking the absolute value of a continuous function is again continuous,

C

⁢

(

X

)

is a sublattice of

ℝ

X

. As a result, we may consider

C

⁢

(

X

)

as a lattice-ordered ring of continuous functions.

Remarks

. Any subring of

C

⁢

(

X

)

is called a

ring of continuous functions

over

X

. This subring may or may not be a sublattice of

C

⁢

(

X

)

. Other than

C

⁢

(

X

)

, the two commonly used lattice-ordered subrings of

C

⁢

(

X

)

are

•

C

*

⁢

(

X

)

, the subset of

C

⁢

(

X

)

consisting of all

bounded

continuous functions. It is easy to see that

C

*

⁢

(

X

)

is closed under all of the algebraic operations (ring-theoretic or lattice-theoretic). So

C

*

⁢

(

X

)

is a lattice-ordered subring of

C

⁢

(

X

)

. When

X

is

pseudocompact

, and in particular, when

X

is

compact

,

C

*

⁢

(

X

)

=

C

⁢

(

X

)

.

In this subring, there is a natural norm that can be defined:

∥

f

∥

:=

sup

x

∈

X

⁡

|

f

⁢

(

x

)

|

=

inf

⁡

{

r

∈

ℝ

∣

|

f

|

≤

r

}

.

Routine verifications show that

∥

f

⁢

g

∥

≤

∥

f

∥

⁢

∥

g

∥

, so that

C

*

⁢

(

X

)

becomes a

normed ring

.

•

The subset of

C

*

⁢

(

X

)

consisting of all constant functions. This is

isomorphic

to

ℝ

, and is often identified as such, so that

ℝ

is considered as a lattice-ordered subring of

C

⁢

(

X

)

.

References

1

L. Gillman, M. Jerison:

Rings of Continuous Functions

, Van Nostrand, (1960).

Title

ring of continuous functions

Canonical name

RingOfContinuousFunctions

Date of creation

2013-03-22 16:54:54

Last modified on

2013-03-22 16:54:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54C40

Classification

msc 54C35

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