Namespaces
Variants
Actions

Difference between revisions of "Functional"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (TeX encoding is done)
 
Line 1: Line 1:
A [[Mapping|mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042010/f0420101.png" /> of an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042010/f0420102.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042010/f0420103.png" /> of real numbers or the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042010/f0420104.png" /> of complex numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042010/f0420105.png" /> is endowed with the structure of a [[Vector space|vector space]], a [[Topological space|topological space]] or an [[Ordered set|ordered set]], then there arise the important classes of linear, continuous and monotone functionals, respectively (cf. [[Linear functional|Linear functional]]; [[Continuous functional|Continuous functional]]; [[Monotone mapping|Monotone mapping]]).
+
{{TEX|done}}
 +
 
 +
A [[Mapping|mapping]] $f$ of an arbitrary set $X$ into the set $\mathbb R$ of real numbers or the set $\mathbb C$ of complex numbers. If $X$ is endowed with the structure of a [[Vector space|vector space]], a [[Topological space|topological space]] or an [[Ordered set|ordered set]], then there arise the important classes of linear, continuous and monotone functionals, respectively (cf. [[Linear functional|Linear functional]]; [[Continuous functional|Continuous functional]]; [[Monotone mapping|Monotone mapping]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>

Latest revision as of 06:47, 16 January 2013


A mapping $f$ of an arbitrary set $X$ into the set $\mathbb R$ of real numbers or the set $\mathbb C$ of complex numbers. If $X$ is endowed with the structure of a vector space, a topological space or an ordered set, then there arise the important classes of linear, continuous and monotone functionals, respectively (cf. Linear functional; Continuous functional; Monotone mapping).

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional&oldid=29314
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article