Namespaces
Variants
Actions

Difference between revisions of "Bounded operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174201.png" /> of a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174202.png" /> into a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174203.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174204.png" /> is a bounded subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174205.png" /> for any bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174207.png" />. Every operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174208.png" />, continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b0174209.png" />, is a bounded operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742010.png" /> is a linear operator, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742011.png" /> to be bounded it is sufficient that there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742013.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742014.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742016.png" /> are normed linear spaces and that the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742017.png" /> is bounded. Then
+
<!--
 +
b0174201.png
 +
$#A+1 = 52 n = 0
 +
$#C+1 = 52 : ~/encyclopedia/old_files/data/B017/B.0107420 Bounded operator
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742018.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
This number is called the norm of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742019.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742020.png" />. Then
+
A mapping  $  A $
 +
of a topological vector space  $  X $
 +
into a topological vector space  $  Y $
 +
such that  $  A (M) $
 +
is a bounded subset in  $  Y $
 +
for any bounded subset  $  M $
 +
of $  X $.
 +
Every operator $  A : X \rightarrow Y $,
 +
continuous on  $  X $,
 +
is a bounded operator. If  $  A :  X \rightarrow Y $
 +
is a linear operator, then for  $  A $
 +
to be bounded it is sufficient that there exists a neighbourhood  $  U \subset  X $
 +
such that  $  A (U) $
 +
is bounded in  $  Y $.  
 +
Suppose that  $  X $
 +
and  $  Y $
 +
are normed linear spaces and that the linear operator  $  A :  X \rightarrow Y $
 +
is bounded. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742021.png" /></td> </tr></table>
+
$$
 +
\gamma  = \
 +
\sup _ {\| x \| \leq  1 }  \| A x \|  < \infty .
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742022.png" /> is the smallest constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742023.png" /> such that
+
This number is called the norm of the operator  $  A $
 +
and is denoted by  $  \| A \| $.  
 +
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742024.png" /></td> </tr></table>
+
$$
 +
\| A x \|  \leq  \| A  \| \cdot \|  x \| ,
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742025.png" />. Conversely, if this inequality is satisfied, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742026.png" /> is bounded. For linear operators mapping a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742027.png" /> into a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742028.png" />, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742030.png" />, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742031.png" /> is bornological and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742032.png" /> is a locally convex space, then the boundedness of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742033.png" /> implies its continuity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742034.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742035.png" /> is a bounded symmetric operator, then the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742036.png" /> is bounded on the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742037.png" />. The numbers
+
and $  \| A \| $
 +
is the smallest constant  $  C $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742038.png" /></td> </tr></table>
+
$$
 +
\| A x \|  \leq  C  \| x \|
 +
$$
  
are called the upper and lower bounds of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742039.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742041.png" /> belongs to the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742042.png" />, and the whole spectrum lies in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742043.png" />. Examples of bounded operators are: the projection operator ([[Projector|projector]]) onto a complemented subspace of a Banach space, and an [[Isometric operator|isometric operator]] acting on a Hilbert space.
+
for any  $  x \in X $.
 +
Conversely, if this inequality is satisfied, then  $  A $
 +
is bounded. For linear operators mapping a normed space  $  X $
 +
into a normed space  $  Y $,
 +
the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces  $  X $
 +
and $  Y $,
 +
but if  $  X $
 +
is bornological and  $  Y $
 +
is a locally convex space, then the boundedness of a linear operator $  A : X \rightarrow Y $
 +
implies its continuity. If  $  H $
 +
is a Hilbert space and $  A : H \rightarrow H $
 +
is a bounded symmetric operator, then the quadratic form  $  \langle  A x , x \rangle $
 +
is bounded on the unit ball  $  K _ {1} = \{ {x } : {\| x \| \leq  1 } \} $.  
 +
The numbers
  
If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742045.png" /> have the structure of a partially ordered set, for example are vector lattices (cf. [[Vector lattice|Vector lattice]]), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742046.png" /> is called order-bounded if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742047.png" /> is an order-bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742048.png" /> for any order-bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742050.png" />. Examples: an isotone operator, i.e. an operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742051.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017420/b01742052.png" />.
+
$$
 +
\beta  = \
 +
\sup _ {\| x \| \leq  1 }  \langle  A x , x \rangle \ \
 +
\textrm{ and } \ \
 +
\alpha  = \inf _ {\| x \| \leq  1 }  \langle  A x , x \rangle
 +
$$
 +
 
 +
are called the upper and lower bounds of the operator  $  A $.  
 +
The points  $  \alpha $
 +
and  $  \beta $
 +
belongs to the spectrum of  $  A $,
 +
and the whole spectrum lies in the interval  $  [ \alpha , \beta ] $.  
 +
Examples of bounded operators are: the projection operator ([[Projector|projector]]) onto a complemented subspace of a Banach space, and an [[Isometric operator|isometric operator]] acting on a Hilbert space.
 +
 
 +
If the space  $  X $
 +
and $  Y $
 +
have the structure of a partially ordered set, for example are vector lattices (cf. [[Vector lattice|Vector lattice]]), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator $  A : X \rightarrow Y $
 +
is called order-bounded if $  A (M) $
 +
is an order-bounded set in $  Y $
 +
for any order-bounded set $  M $
 +
in $  X $.  
 +
Examples: an isotone operator, i.e. an operator such that $  x \leq  y $
 +
implies $  A x \leq  A y $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elemente der Funktionalanalysis" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elemente der Funktionalanalysis" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====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><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


A mapping $ A $ of a topological vector space $ X $ into a topological vector space $ Y $ such that $ A (M) $ is a bounded subset in $ Y $ for any bounded subset $ M $ of $ X $. Every operator $ A : X \rightarrow Y $, continuous on $ X $, is a bounded operator. If $ A : X \rightarrow Y $ is a linear operator, then for $ A $ to be bounded it is sufficient that there exists a neighbourhood $ U \subset X $ such that $ A (U) $ is bounded in $ Y $. Suppose that $ X $ and $ Y $ are normed linear spaces and that the linear operator $ A : X \rightarrow Y $ is bounded. Then

$$ \gamma = \ \sup _ {\| x \| \leq 1 } \| A x \| < \infty . $$

This number is called the norm of the operator $ A $ and is denoted by $ \| A \| $. Then

$$ \| A x \| \leq \| A \| \cdot \| x \| , $$

and $ \| A \| $ is the smallest constant $ C $ such that

$$ \| A x \| \leq C \| x \| $$

for any $ x \in X $. Conversely, if this inequality is satisfied, then $ A $ is bounded. For linear operators mapping a normed space $ X $ into a normed space $ Y $, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces $ X $ and $ Y $, but if $ X $ is bornological and $ Y $ is a locally convex space, then the boundedness of a linear operator $ A : X \rightarrow Y $ implies its continuity. If $ H $ is a Hilbert space and $ A : H \rightarrow H $ is a bounded symmetric operator, then the quadratic form $ \langle A x , x \rangle $ is bounded on the unit ball $ K _ {1} = \{ {x } : {\| x \| \leq 1 } \} $. The numbers

$$ \beta = \ \sup _ {\| x \| \leq 1 } \langle A x , x \rangle \ \ \textrm{ and } \ \ \alpha = \inf _ {\| x \| \leq 1 } \langle A x , x \rangle $$

are called the upper and lower bounds of the operator $ A $. The points $ \alpha $ and $ \beta $ belongs to the spectrum of $ A $, and the whole spectrum lies in the interval $ [ \alpha , \beta ] $. Examples of bounded operators are: the projection operator (projector) onto a complemented subspace of a Banach space, and an isometric operator acting on a Hilbert space.

If the space $ X $ and $ Y $ have the structure of a partially ordered set, for example are vector lattices (cf. Vector lattice), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator $ A : X \rightarrow Y $ is called order-bounded if $ A (M) $ is an order-bounded set in $ Y $ for any order-bounded set $ M $ in $ X $. Examples: an isotone operator, i.e. an operator such that $ x \leq y $ implies $ A x \leq A y $.

References

[1] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1955) (Translated from Russian)
[2] W. Rudin, "Functional analysis" , McGraw-Hill (1973)
[3] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)

Comments

References

[a1] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
[a2] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
How to Cite This Entry:
Bounded operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bounded_operator&oldid=46139
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article