function space

function space

Generally speaking, a

function space

is a

collection

of

functions

satisfying certain properties. Typically, these properties are topological in nature, and hence the word “space”. Usually, functions in a function space have a common domain (

http://planetmath.org/Function

) and

codomain

. Thus, a function space

ℱ

, which contains functions acting from set

X

to set

Y

, is denoted by

ℱ

⁢

(

X

,

Y

)

. Evidently,

ℱ

⁢

(

X

,

Y

)

⊆

Y

X

. In the case when

Y

=

ℝ

one usually writes only

ℱ

⁢

(

X

)

.

If the codomain

Y

is a vector space over field

K

, then it is easy to define

operations

of the vector space on functions acting to

Y

in the following way:

(

α

⋅

f

)

⁢

(

x

)

=

α

⋅

f

⁢

(

x

)

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

(1)

where

α

is an

element of

the field

K

, and

x

is an element of the domain (

http://planetmath.org/Function

) of functions.
One usually consider function spaces which are closed under operations (

1

) and thus
are vector spaces. Function spaces are also often equipped with some

topology

.

Below is a list of function spaces, to entries where they are defined, and notation for these.

The main purpose of this entry is to give a list
of function spaces that already have been
defined on PlanetMath (or should be), a gallery of function
spaces if you like.

Restrictions on smoothness

•

C

;

continuous functions

•

C

k

;

k

times

continuously differentiable functions

•

C

k

,

α

; Hölder continuous functions

•

Lip

;

Lipschitz continuous

functions

•

C

∞

;

smooth functions

•

C

ω

;

analytic functions

•

𝒪

⁢

(

G

)

;

holomorphic functions

•

C

c

∞

or

𝒟

;

smooth functions with compact support

Restrictions on integrability

•

L

0

;

measurable functions

•

L

1

; integrable functions

•

L

2

;

square integrable

functions

•

L

p

functions (

http://planetmath.org/LpSpace

)

•

L

∞

;

essentially bounded functions

•

L

loc

1

⁢

(

U

)

;

locally integrable function

Integrability of derivatives

•

B

⁢

V

;

functions of bounded variation

, i.e. functions whose

derivative

is a

measure

•

W

m

,

p

⁢

(

Ω

)

;

Sobolev space

of

p

-integrable functions which have

p

-

integrable

derivatives of

m

-th order. Space

W

m

,

2

⁢

(

Ω

)

is a

Hilbert space

and is usually denoted by

W

m

⁢

(

Ω

)

or

H

m

⁢

(

Ω

)

.

•

B

⁢

M

⁢

O

; functions with bounded mean oscillation.

V

⁢

M

⁢

O

functions with vanishing mean

oscillation

Restriction on growth

•

B

;

bounded functions

•

Functions with polynomial growth

•

𝒮

; rapidly

decreasing functions

(

Schwartz space

)

Test function spaces

•

𝒮

; rapidly decreasing functions (Schwartz space)

•

𝒟

; smooth functions with compact support

Distribution spaces

•

𝒮

′

;

tempered distributions

•

𝒟

′

;

distributions

•

ℰ

′

; distributions with compact support

•

ℳ

;

Radon measures

Piecewise properties

•

P

⁢

C

;

piecewise continuous

functions

•

P

⁢

C

k

;

piecewise

k times

continuous

differentiable functions

•

P

⁢

C

∞

; piecewise smooth functions (

http://planetmath.org/PiecewiseSmooth

)

•

piecewise linear functions

•

simple functions

It is possible to attach a number which we call

regularity

index

, to many of these spaces.
If a space

X

has a regularity index which is strictly less than the regularity index of

Y

, then (under some

hypothesis

on the domain of the functions)

X

contains

Y

.

Here is a list of regularity indices (

n

is the

dimension

of the domain):

C

0

C

k

k

C

∞

∞

C

ω

∞

C

k

,

α

k

+

α

Lip

1

L

p

-

n

/

p

L

∞

0

W

k

,

p

k

-

n

/

p

W

k

,

∞

k

B

⁢

V

0

𝒟

′

-

∞

ℳ

-

n

Selected links

•

The entry

\htmladdnormallink

Function spacehttp://en.wikipedia.org/wiki/Function_space
at the

\htmladdnormallink

Wikipediahttp://en.wikipedia.org/.

•

Chapter

\htmladdnormallink

Function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/node2.html
from the

\htmladdnormallink

Notes on distributions and function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/
by

\htmladdnormallink

D. Stewarthttp://www.math.uiowa.edu/ dstewart/.

Title

function space

Canonical name

FunctionSpace

Date of creation

2013-03-22 14:08:31

Last modified on

2013-03-22 14:08:31

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

38

Author

matte (1858)

Entry type

Topic

Classification

msc 54C35

Classification

msc 26-00

Classification

msc 46-00

Classification

msc 30H05

Synonym

space of functions

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