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Adjoint norms (cf. [[Norm|Norm]]) in certain vector spaces dual to each other.
 
Adjoint norms (cf. [[Norm|Norm]]) in certain vector spaces dual to each other.
  
1) The mass of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m0625902.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m0625903.png" />, i.e. an element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m0625904.png" />-fold exterior product of a vector space, is the number
+
1) The mass of an $  r $-
 +
vector $  \alpha $,  
 +
i.e. an element of the $  r $-
 +
fold exterior product of a vector space, is the number
  
<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/m/m062/m062590/m0625905.png" /></td> </tr></table>
+
$$
 +
| \alpha | _ {0}  = \
 +
\inf \
 +
\left \{ {\sum _ { i } | \alpha _ {i} | } : {\alpha = \sum
 +
{\alpha _ {i} } ,\
 +
\alpha _ {i} \
 +
\textrm{ simple }  r \textrm{ - vectors  } } \right \}
 +
.
 +
$$
  
The co-mass of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m0625907.png" />-covector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m0625908.png" /> is the number
+
The co-mass of an $  r $-
 +
covector $  \omega $
 +
is the number
  
<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/m/m062/m062590/m0625909.png" /></td> </tr></table>
+
$$
 +
| \omega | _ {0}  = \
 +
\sup _  \alpha
 +
\{ {| \omega \cdot \alpha | } : {
 +
\alpha  \textrm{ a  simple  }  r
 +
\textrm{ - vector  } , | \alpha | = 1 } \}
 +
.
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259010.png" /> is the standard norm of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259011.png" />-vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259012.png" /> is the scalar product of a vector and a covector.
+
Here $  | \cdot | $
 +
is the standard norm of an $  r $-
 +
vector and $  \omega \cdot \alpha $
 +
is the scalar product of a vector and a covector.
  
The mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259013.png" /> and the co-mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259014.png" /> are adjoint norms in the spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259015.png" />-vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259017.png" />-covectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259018.png" />, respectively. In this connection:
+
The mass $  | \alpha | _ {0} $
 +
and the co-mass $  | \omega | _ {0} $
 +
are adjoint norms in the spaces of $  r $-
 +
vectors $  V _ {[} r] $
 +
and $  r $-
 +
covectors $  V  ^ {[} r] $,  
 +
respectively. In this connection:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259020.png" />;
+
a) $  | \omega | _ {0} = \sup _  \alpha  \{ {| \omega \cdot \alpha | } : {| \alpha | _ {0} = 1 } \} $,
 +
$  | \alpha | _ {0} = \sup _  \alpha  \{ {| \omega \cdot \alpha | } : {| \omega | _ {0} = 1 } \} $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259022.png" />, and equalities hold if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259024.png" />) is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259025.png" />-(co)vector;
+
b) $  | \alpha | _ {0} \geq  | \alpha | $,  
 +
$  | \omega | _ {0} \geq  | \omega | $,  
 +
and equalities hold if and only if $  \alpha $(
 +
$  \omega $)  
 +
is a simple $  r $-(
 +
co)vector;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259027.png" /> for exterior products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259028.png" />, where for a simple multi-covector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259029.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259030.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259031.png" />, and, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259032.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259034.png" />;
+
c) $  | \alpha \lor \beta | _ {0} \leq  | \alpha | _ {0} | \beta | _ {0} $,  
 +
$  | \omega \lor \zeta | _ {0} \leq  B  | \omega | _ {0} | \zeta | _ {0} $
 +
for exterior products $  \lor $,  
 +
where for a simple multi-covector $  \omega $(
 +
or $  \zeta $)  
 +
$  B = 1 $,  
 +
and, in general, $  B = ( _ {\  r }  ^ {r+} s ) $
 +
if $  \omega \in V  ^ {[} r] $
 +
and $  \zeta \in V  ^ {[} s] $;
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259035.png" /> for inner products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259039.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259042.png" />.
+
d) $  | \omega \wedge \alpha | _ {0} \leq  \widetilde{B}  | \omega | _ {0} | \alpha | _ {0} $
 +
for inner products $  \wedge $,  
 +
where $  \widetilde{B}  = 1 $
 +
for $  r \geq  s $
 +
and $  \widetilde{B}  = ( _ {r}  ^ {s} ) $
 +
for $  r \leq  s $,  
 +
$  \omega \in V  ^ {[} r] $
 +
and $  \alpha \in V _ {[} s] $.
  
These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259044.png" />. For example, the co-mass of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259045.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259046.png" /> is
+
These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are $  V  ^ {[} r] $
 +
and $  V _ {[} r] $.  
 +
For example, the co-mass of a form $  \omega $
 +
on a domain $  G \subset  E  ^ {n} $
 +
is
  
<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/m/m062/m062590/m06259047.png" /></td> </tr></table>
+
$$
 +
| \omega | _ {0}  = \
 +
\sup  \{ {| \omega ( p) | _ {0} } : {p \in G } \}
 +
.
 +
$$
  
2) The mass of a polyhedral chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259048.png" /> is
+
2) The mass of a polyhedral chain $  A = \sum {a _ {i} } \sigma _ {i}  ^ {r} $
 +
is
  
<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/m/m062/m062590/m06259049.png" /></td> </tr></table>
+
$$
 +
| A |  = \sum | a _ {i} |  | \sigma _ {i}  ^ {r} | ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259050.png" /> is the volume of the cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259051.png" />. For arbitrary chains the mass (finite or infinite) can be defined in various ways; for flat chains (see [[Flat norm|Flat norm]]) and sharp chains (see [[Sharp norm|Sharp norm]]) these give the same value to the mass.
+
where $  | \sigma _ {i}  ^ {r} | $
 +
is the volume of the cell $  \sigma _ {i}  ^ {r} $.  
 +
For arbitrary chains the mass (finite or infinite) can be defined in various ways; for flat chains (see [[Flat norm|Flat norm]]) and sharp chains (see [[Sharp norm|Sharp norm]]) these give the same value to the mass.
  
3) The co-mass of a (flat, in particular, sharp) cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259052.png" /> is defined in the standard way:
+
3) The co-mass of a (flat, in particular, sharp) cochain $  X $
 +
is defined in the standard way:
  
<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/m/m062/m062590/m06259053.png" /></td> </tr></table>
+
$$
 +
| X |  = \
 +
\sup _ {A \neq 0 } \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259054.png" /> is a polyhedral chain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259055.png" /> is the value of the cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259056.png" /> on the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259057.png" />.
+
\frac{| X \cdot A | }{| A | }
 +
,
 +
$$
 +
 
 +
where $  A $
 +
is a polyhedral chain and $  X \cdot A $
 +
is the value of the cochain $  X $
 +
on the chain $  A $.
  
 
For references see [[Flat norm|Flat norm]].
 
For references see [[Flat norm|Flat norm]].
 
 
  
 
====Comments====
 
====Comments====
A simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259059.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259060.png" /> is an element of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259061.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259062.png" />-fold [[Exterior product|exterior product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259063.png" /> of a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259064.png" />. Here  ""  denotes exterior product and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062590/m06259065.png" />.
+
A simple $  r $-
 +
vector $  \alpha $
 +
is an element of the form $  \alpha = \beta _ {1} \lor \dots \lor \beta _ {r} $
 +
in the $  r $-
 +
fold [[Exterior product|exterior product]] $  V _ {[} r] $
 +
of a [[Vector space|vector space]] $  V $.  
 +
Here  ""  denotes exterior product and $  \beta _ {1} \dots \beta _ {r} \in V $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)  pp. Sect. 1.8</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)  pp. Sect. 1.8 {{MR|0257325}} {{ZBL|0176.00801}} </TD></TR></table>

Latest revision as of 07:59, 6 June 2020


Adjoint norms (cf. Norm) in certain vector spaces dual to each other.

1) The mass of an $ r $- vector $ \alpha $, i.e. an element of the $ r $- fold exterior product of a vector space, is the number

$$ | \alpha | _ {0} = \ \inf \ \left \{ {\sum _ { i } | \alpha _ {i} | } : {\alpha = \sum {\alpha _ {i} } ,\ \alpha _ {i} \ \textrm{ simple } r \textrm{ - vectors } } \right \} . $$

The co-mass of an $ r $- covector $ \omega $ is the number

$$ | \omega | _ {0} = \ \sup _ \alpha \{ {| \omega \cdot \alpha | } : { \alpha \textrm{ a simple } r \textrm{ - vector } , | \alpha | = 1 } \} . $$

Here $ | \cdot | $ is the standard norm of an $ r $- vector and $ \omega \cdot \alpha $ is the scalar product of a vector and a covector.

The mass $ | \alpha | _ {0} $ and the co-mass $ | \omega | _ {0} $ are adjoint norms in the spaces of $ r $- vectors $ V _ {[} r] $ and $ r $- covectors $ V ^ {[} r] $, respectively. In this connection:

a) $ | \omega | _ {0} = \sup _ \alpha \{ {| \omega \cdot \alpha | } : {| \alpha | _ {0} = 1 } \} $, $ | \alpha | _ {0} = \sup _ \alpha \{ {| \omega \cdot \alpha | } : {| \omega | _ {0} = 1 } \} $;

b) $ | \alpha | _ {0} \geq | \alpha | $, $ | \omega | _ {0} \geq | \omega | $, and equalities hold if and only if $ \alpha $( $ \omega $) is a simple $ r $-( co)vector;

c) $ | \alpha \lor \beta | _ {0} \leq | \alpha | _ {0} | \beta | _ {0} $, $ | \omega \lor \zeta | _ {0} \leq B | \omega | _ {0} | \zeta | _ {0} $ for exterior products $ \lor $, where for a simple multi-covector $ \omega $( or $ \zeta $) $ B = 1 $, and, in general, $ B = ( _ {\ r } ^ {r+} s ) $ if $ \omega \in V ^ {[} r] $ and $ \zeta \in V ^ {[} s] $;

d) $ | \omega \wedge \alpha | _ {0} \leq \widetilde{B} | \omega | _ {0} | \alpha | _ {0} $ for inner products $ \wedge $, where $ \widetilde{B} = 1 $ for $ r \geq s $ and $ \widetilde{B} = ( _ {r} ^ {s} ) $ for $ r \leq s $, $ \omega \in V ^ {[} r] $ and $ \alpha \in V _ {[} s] $.

These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are $ V ^ {[} r] $ and $ V _ {[} r] $. For example, the co-mass of a form $ \omega $ on a domain $ G \subset E ^ {n} $ is

$$ | \omega | _ {0} = \ \sup \{ {| \omega ( p) | _ {0} } : {p \in G } \} . $$

2) The mass of a polyhedral chain $ A = \sum {a _ {i} } \sigma _ {i} ^ {r} $ is

$$ | A | = \sum | a _ {i} | | \sigma _ {i} ^ {r} | , $$

where $ | \sigma _ {i} ^ {r} | $ is the volume of the cell $ \sigma _ {i} ^ {r} $. For arbitrary chains the mass (finite or infinite) can be defined in various ways; for flat chains (see Flat norm) and sharp chains (see Sharp norm) these give the same value to the mass.

3) The co-mass of a (flat, in particular, sharp) cochain $ X $ is defined in the standard way:

$$ | X | = \ \sup _ {A \neq 0 } \ \frac{| X \cdot A | }{| A | } , $$

where $ A $ is a polyhedral chain and $ X \cdot A $ is the value of the cochain $ X $ on the chain $ A $.

For references see Flat norm.

Comments

A simple $ r $- vector $ \alpha $ is an element of the form $ \alpha = \beta _ {1} \lor \dots \lor \beta _ {r} $ in the $ r $- fold exterior product $ V _ {[} r] $ of a vector space $ V $. Here "" denotes exterior product and $ \beta _ {1} \dots \beta _ {r} \in V $.

References

[a1] H. Federer, "Geometric measure theory" , Springer (1969) pp. Sect. 1.8 MR0257325 Zbl 0176.00801
How to Cite This Entry:
Mass and co-mass. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mass_and_co-mass&oldid=13071
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article