Function Space Article Index for
Function
Website Links For
Function
 

Information About

Function Space




  • in Set Theory , the Power Set of a set ''X'' may be identified with the set of all functions from ''X'' to {0,1};, denoted 2''X''. More generally, the set of functions ''X'' → ''Y'' is denoted ''Y''''X''.





  • in Topology , one may attempt to put a topology on the space of continuous functions from a Topological Space ''X'' to another one ''Y'', with utility depending on the nature of the spaces. A commonly used example is the Compact-open Topology . Also available is the Product Topology on the space of set theoretic functions (i.e. not necessarily continuous functions) ''Y''''X''. In this context, this topology is also referred to as the Topology Of Pointwise Convergence .








LIST OF FUNCTION SPACES


Functional analysis



Abstract spaces

  • s (equivalently, a local base of convex sets).

  • collection of seminorms (equivalently, a translation invariant metric).

  • collection of seminorms (equivalently, a single Norm ).

  • .



Concrete spaces


  • Schwartz Space of smooth functions of rapid decrease and its dual, tempered distributions

  • Lp Space

  • κ(R) continuous compact support with uniform norm

  • ''B''(R) bounded continuous ( Bounded Function )

  • ''C''(R) functions which vanish at infinity

  • ''C''(R) Smooth Functions

  • ''C''0 smooth compact support uniform norm (and the one with derivatives)

  • ''D''(R) compact support in limit topology

  • ''W''''k'',''p'' Sobolev Space

  • O''U'' holomorphic functions

  • linear functions

  • piecewise linear functions

  • continuous functions, compact open topology

  • all functions, space of pointwise convergence

  • Hardy Space

  • Hölder Space



SEE ALSO