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Difference between revisions of "Continuous functional"

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A [[Continuous operator|continuous operator]] ([[Continuous mapping|continuous mapping]]) mapping a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256601.png" />, which as a rule is also a vector space, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256602.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256603.png" />. Therefore, the definition of, and criteria for, continuity of an arbitrary operator continue to hold for functionals. For example,
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A [[continuous operator]] ([[continuous mapping]]) mapping a topological space $X$, which as a rule is also a vector space, into $\mathbf{R}$ or $\mathbf{C}$. Therefore, the definition of, and criteria for, continuity of an arbitrary operator continue to hold for functionals. For example,
  
1) for a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256604.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256605.png" /> is a subset of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256606.png" />, to be continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256607.png" /> there must for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256608.png" /> be a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c0256609.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566012.png" /> (definition of continuity of functionals);
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1) for a functional $f : M \rightarrow \mathbf{C}$, where $M$ is a subset of a topological space $X$, to be continuous at a point $x_0 \in M$ there must for any $\epsilon > 0$ be a neighbourhood $U$ of $x_0$ such that $|f(x) - f(x_0| < \epsilon$ for $x \in U$ (definition of continuity of functionals);
  
 
2) a functional that is continuous on a compact set of a separable topological vector space is bounded on this set and attains its least upper and greatest lower bounds (Weierstrass' theorem);
 
2) a functional that is continuous on a compact set of a separable topological vector space is bounded on this set and attains its least upper and greatest lower bounds (Weierstrass' theorem);
  
3) since every non-zero linear functional maps a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566013.png" /> onto the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566014.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566015.png" />), it induces an open mapping, that is, the image of any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566016.png" /> is an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566017.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025660/c02566018.png" />).
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3) since every non-zero linear functional maps a Banach space $X$ onto the whole of $\mathbf{R}$ (or $\mathbf{C}$), it induces an open mapping, that is, the image of any open set $G \subset X$ is an open set in $\mathbf{R}$ (or $\mathbf{C}$).
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR>
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</table>
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Revision as of 21:23, 20 January 2017

A continuous operator (continuous mapping) mapping a topological space $X$, which as a rule is also a vector space, into $\mathbf{R}$ or $\mathbf{C}$. Therefore, the definition of, and criteria for, continuity of an arbitrary operator continue to hold for functionals. For example,

1) for a functional $f : M \rightarrow \mathbf{C}$, where $M$ is a subset of a topological space $X$, to be continuous at a point $x_0 \in M$ there must for any $\epsilon > 0$ be a neighbourhood $U$ of $x_0$ such that $|f(x) - f(x_0| < \epsilon$ for $x \in U$ (definition of continuity of functionals);

2) a functional that is continuous on a compact set of a separable topological vector space is bounded on this set and attains its least upper and greatest lower bounds (Weierstrass' theorem);

3) since every non-zero linear functional maps a Banach space $X$ onto the whole of $\mathbf{R}$ (or $\mathbf{C}$), it induces an open mapping, that is, the image of any open set $G \subset X$ is an open set in $\mathbf{R}$ (or $\mathbf{C}$).


Comments

References

[a1] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
How to Cite This Entry:
Continuous functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_functional&oldid=40193
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article