Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » An Introduction to Wavelet Analysis » Function spaces: notion and notations

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • IEEE-SPS

    This collection is included inLens: IEEE Signal Processing Society Lens
    By: IEEE Signal Processing Society

    Click the "IEEE-SPS" link to see all content they endorse.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • NSF Partnership display tagshide tags

    This collection is included inLens: NSF Partnership in Signal Processing
    By: Sidney Burrus

    Click the "NSF Partnership" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • UniqU content

    This collection is included inLens: UniqU's lens
    By: UniqU, LLC

    Click the "UniqU content" link to see all content selected in this lens.

  • Lens for Engineering

    This module and collection are included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Function spaces: notion and notations

Module by: Veronique Delouille. E-mail the author

A Hilbert space is a complete normed space whose norm is indexed by an inner (or scalar) product.

Two disjoint subspaces AA and BB of a space SS form a direct sum decomposition of SS if every element of SS can be written uniquely as a sum of an element of AA and an element of BB. The notation S=ABS=AB is then used.

A measurable function ff belongs to the Lebesgue space Lp(R),1p<Lp(R),1p< if

f p = - + f ( x ) p 1 / p < . f p = - + f ( x ) p 1 / p < .
(1)

An example of a Hilbert space is the Lebesgue space L2(R)L2(R) of measurable and square integrable functions. Indeed, the norm ·2·2 is induced by the scalar product

f , g = f ( x ) g ( x ) ¯ d x , f , g = f ( x ) g ( x ) ¯ d x ,
(2)

where g(x)¯g(x)¯ denotes the complex conjugate of g(x)g(x). Two functions are said to be orthogonal in L2(R)L2(R) if their inner product is zero.

The Lebesgue measure can be replaced by a more general measure μμ, leading to the weighted space L2(μ)L2(μ), which has as inner product

f , g d μ = f ( x ) g ( x ) ¯ d μ ( x ) f , g d μ = f ( x ) g ( x ) ¯ d μ ( x )
(3)

and which contains the functions that have a finite norm fdμ:=f,fdμ<fdμ:=f,fdμ<.

A countable subset {fk}{fk} of functions belonging to a Hilbert space is a Riesz basis if every element ff of the space can be written uniquely as f=kckfkf=kckfk, and if positive constants AA and BB exist such that

A f 2 2 k c k 2 B f 2 2 . A f 2 2 k c k 2 B f 2 2 .
(4)

A Riesz basis is an orthogonal basis if the fkfk are mutually orthogonal. In this case, A=B=1A=B=1.

References

    Collection Navigation

    Content actions

    Download:

    Collection as:

    PDF | EPUB (?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

    | More downloads ...

    Module as:

    PDF | More downloads ...

    Add:

    Collection to:

    My Favorites (?)

    'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

    | A lens I own (?)

    Definition of a lens

    Lenses

    A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

    What is in a lens?

    Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

    Who can create a lens?

    Any individual member, a community, or a respected organization.

    What are tags? tag icon

    Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    | External bookmarks

    Module to:

    My Favorites (?)

    'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

    | A lens I own (?)

    Definition of a lens

    Lenses

    A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

    What is in a lens?

    Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

    Who can create a lens?

    Any individual member, a community, or a respected organization.

    What are tags? tag icon

    Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    | External bookmarks