Namespaces
Variants
Actions

Difference between revisions of "Sharp norm"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (fix tex)
Line 24: Line 24:
  
 
$$  
 
$$  
{| \partial  \sigma  ^ {r+} 1 | }  ^  \prime  \leq  | \sigma  ^ {r+} 1 | ,
+
{| \partial  \sigma  ^ {r+1} | }  ^  \prime  \leq  | \sigma  ^ {r+1} | ,
 
$$
 
$$
  
Line 30: Line 30:
 
{| T _ {v} \sigma  ^ {r} - \sigma  ^ {r} | }  ^  \prime  \leq   
 
{| T _ {v} \sigma  ^ {r} - \sigma  ^ {r} | }  ^  \prime  \leq   
 
\frac{
 
\frac{
{| \sigma  ^ {r} | }  ^  \prime  | v | }{r+}
+
{| \sigma  ^ {r} | }  ^  \prime  | v | }{r+1} ,
1 ,
 
 
$$
 
$$
  
Line 39: Line 38:
  
 
If  $  A = \sum a _ {i} \sigma _ {i}  ^ {r} $,  
 
If  $  A = \sum a _ {i} \sigma _ {i}  ^ {r} $,  
the sharp norm  $  | A |  ^  \srp $
+
the sharp norm  $  | A |  ^  \sharp $
 
is expressed as follows:
 
is expressed as follows:
  
 
$$  
 
$$  
| A |  ^  \srp   =  \inf \left \{  
+
| A |  ^  \sharp   =  \inf \left \{  
\frac{\sum | a _ {i} |  | \sigma _ {i}  ^ {r} |  | v _ {i} | }{r+}
+
\frac{\sum | a _ {i} |  | \sigma _ {i}  ^ {r} |  | v _ {i} | }{r+1} + \left |
1 + \left |
+
\sum a _ {i} T _ {v _ {i}  } \sigma _ {i}  ^ {r} \right |  ^  \flat \right \} ,
\sum a _ {i} T _ {v _ {i}  } \sigma _ {i}  ^ {r} \right |  ^  \flt \right \} ,
 
 
$$
 
$$
  
where  $  | C |  ^  \flt $
+
where  $  | C |  ^  \flat $
 
is the [[Flat norm|flat norm]] of the chain  $  C $,  
 
is the [[Flat norm|flat norm]] of the chain  $  C $,  
 
and the infimum is taken over all shifts  $  v $.
 
and the infimum is taken over all shifts  $  v $.
Line 56: Line 54:
  
 
$$  
 
$$  
| a A |  ^  \srp   =  | a |  | A |  ^  \srp ,
+
| a A |  ^  \sharp   =  | a |  | A |  ^  \sharp ,
 
$$
 
$$
  
 
$$  
 
$$  
| A + B |  ^  \srp   \leq  | A |  ^  \srp + | B |  ^  \srp ,
+
| A + B |  ^  \sharp   \leq  | A |  ^  \sharp + | B |  ^  \sharp ,
 
$$
 
$$
  
 
$$  
 
$$  
| A |  ^  \srp = 0  \iff  A = 0 ,
+
| A |  ^  \sharp = 0  \iff  A = 0 ,
 
$$
 
$$
  
 
$$  
 
$$  
| A |  ^  \srp   \leq  | A |  ^  \flt ;
+
| A |  ^  \sharp   \leq  | A |  ^  \flat ;
 
$$
 
$$
  
 
if  $  r = 0 $,  
 
if  $  r = 0 $,  
then  $  | A |  ^  \srp = | A |  ^  \flt $.
+
then  $  | A |  ^  \sharp = | A |  ^  \flat $.
  
 
The completion of the space  $  C _ {r} ( E  ^ {n} ) $
 
The completion of the space  $  C _ {r} ( E  ^ {n} ) $
is the separable Banach space  $  C _ {r}  ^  \srp ( E  ^ {n} ) $,  
+
is the separable Banach space  $  C _ {r}  ^  \sharp ( E  ^ {n} ) $,  
whose elements are known as  $  r $-
+
whose elements are known as  $  r $-dimensional sharp chains. For any  $  r $-dimensional polyhedral chain  $  A $
dimensional sharp chains. For any  $  r $-
 
dimensional polyhedral chain  $  A $
 
 
and any vector  $  v $,
 
and any vector  $  v $,
  
 
$$  
 
$$  
| T _ {v} A - A |  ^  \srp   \leq   
+
| T _ {v} A - A |  ^  \sharp   \leq   
\frac{| A |  | v | }{r+}
+
\frac{| A |  | v | }{r+1} ,
1 ,
 
 
$$
 
$$
  
Line 91: Line 86:
 
by the vector  $  v $
 
by the vector  $  v $
 
of length  $  | v | $.  
 
of length  $  | v | $.  
A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If  $  A = \lim\limits  ^  \flt   A _ {i} $,  
+
A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If  $  A = \lim\limits  ^  \flat   A _ {i} $,  
 
where  $  A _ {i} $
 
where  $  A _ {i} $
are polyhedral chains, and  $  \psi A = \lim\limits  ^  \srp   A _ {i} $,  
+
are polyhedral chains, and  $  \psi A = \lim\limits  ^  \sharp   A _ {i} $,  
 
where  $  \psi $
 
where  $  \psi $
is a linear bijective mapping from the space  $  C _ {r}  ^  \flt ( E  ^ {n)} $
+
is a linear bijective mapping from the space  $  C _ {r}  ^  \flat ( E  ^ {n)} $
into the space  $  C _ {r}  ^  \srp ( E  ^ {n)} $,  
+
into the space  $  C _ {r}  ^  \sharp ( E  ^ {n)} $,  
and  $  \psi C _ {r}  ^  \flt $
+
and  $  \psi C _ {r}  ^  \flat $
is dense in  $  C _ {r}  ^  \srp $
+
is dense in  $  C _ {r}  ^  \sharp $
 
in the sharp norm.
 
in the sharp norm.
  
Line 104: Line 99:
 
of a sharp chain [[#References|[1]]]; an  $  r $-
 
of a sharp chain [[#References|[1]]]; an  $  r $-
 
dimensional sharp chain  $  X = XA $
 
dimensional sharp chain  $  X = XA $
is an element of the space  $  C ^ {\srp r } ( E  ^ {n} ) $
+
is an element of the space  $  C ^ {\sharp r } ( E  ^ {n} ) $
dual to  $  C _ {r}  ^  \srp ( E  ^ {n} ) $;  
+
dual to  $  C _ {r}  ^  \sharp ( E  ^ {n} ) $;  
 
it is a flat cochain, and
 
it is a flat cochain, and
  
 
$$  
 
$$  
| X |  \leq  | X |  ^  \flt   \leq  | X |  ^  \srp ,
+
| X |  \leq  | X |  ^  \flat   \leq  | X |  ^  \sharp ,
 
$$
 
$$
  
 
where  $  | X | $
 
where  $  | X | $
 
is the co-mass of  $  X $,  
 
is the co-mass of  $  X $,  
while the sharp co-norm  $  | X |  ^  \srp $
+
while the sharp co-norm  $  | X |  ^  \sharp $
is defined similarly to the flat norm  $  | X |  ^  \flt $.  
+
is defined similarly to the flat norm  $  | X |  ^  \flat $.  
 
The co-boundary  $  dX $
 
The co-boundary  $  dX $
 
of a sharp cochain is not necessarily sharp [[#References|[1]]], but
 
of a sharp cochain is not necessarily sharp [[#References|[1]]], but
  
 
$$  
 
$$  
| dX |  \leq  | X |  ^  \flt   \leq  | X |  ^  \srp .
+
| dX |  \leq  | X |  ^  \flat   \leq  | X |  ^  \sharp .
 
$$
 
$$
  
Line 138: Line 133:
  
 
$$  
 
$$  
( r + 1 ) {\mathcal L} ( X)  \leq  | X |  ^  \srp .
+
( r + 1 ) {\mathcal L} ( X)  \leq  | X |  ^  \sharp .
 
$$
 
$$
  
Line 144: Line 139:
  
 
$$  
 
$$  
| X |  ^  \srp   =  \sup \{ | X |  ^  \flt ,\  
+
| X |  ^  \sharp   =  \sup \{ | X |  ^  \flat ,\  
 
( r + 1 ) {\mathcal L} ( X) \} ,
 
( r + 1 ) {\mathcal L} ( X) \} ,
 
$$
 
$$
Line 170: Line 165:
 
| T _ {v} \gamma - \gamma |  ^  \prime  \leq  \  
 
| T _ {v} \gamma - \gamma |  ^  \prime  \leq  \  
  
\frac{| v |  | \gamma | }{r+}
+
\frac{| v |  | \gamma | }{r+1} ,
1 ,
 
 
$$
 
$$
  
where  $  T _ {v} \gamma ( Q) = \gamma T _ {-} v ( Q) $
+
where  $  T _ {v} \gamma ( Q) = \gamma T _ {-v} ( Q) $
 
is the shift of the function  $  \gamma $
 
is the shift of the function  $  \gamma $
 
by the vector  $  v $
 
by the vector  $  v $
Line 180: Line 174:
  
 
$$  
 
$$  
T _ {-} v ( Q)  =  \{ {q - v } : {q \in Q \subset  E  ^ {n} } \}
+
T _ {-v} ( Q)  =  \{ {q - v } : {q \in Q \subset  E  ^ {n} } \}
 
;
 
;
 
$$
 
$$
Line 189: Line 183:
 
such that  $  | \gamma |  ^  \prime  \leq  \epsilon | \gamma | $
 
such that  $  | \gamma |  ^  \prime  \leq  \epsilon | \gamma | $
 
if the support  $  \supp  t \gamma \subset  U _  \eta  ( p) $
 
if the support  $  \supp  t \gamma \subset  U _  \eta  ( p) $
and  $  \gamma ( E  ^ {n)} = 0 $.
+
and  $  \gamma ( E  ^ {n}) = 0 $.
  
The sharp norm  $  | \gamma |  ^  \srp $
+
The sharp norm  $  | \gamma |  ^  \sharp $
 
is represented as follows:
 
is represented as follows:
  
 
$$  
 
$$  
| \gamma |  ^  \srp   =  \sup _  \omega  \int\limits _ {E  ^ {n}
+
| \gamma |  ^  \sharp   =  \sup _  \omega  \int\limits _ {E  ^ {n}
 
  } \omega  d \gamma ,
 
  } \omega  d \gamma ,
 
$$
 
$$
Line 201: Line 195:
 
where the  $  \omega $
 
where the  $  \omega $
 
are  $  r $-
 
are  $  r $-
dimensional sharp forms for which  $  | \omega |  ^  \srp \leq  1 $.
+
dimensional sharp forms for which  $  | \omega |  ^  \sharp \leq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)</TD></TR></table>

Revision as of 21:46, 20 February 2021


in the space of $ r $- dimensional polyhedral chains $ C _ {r} ( E ^ {n)} $

The largest semi-norm $ {| \cdot | } ^ \prime $ which, for any cell $ \sigma ^ {r} $ of volume $ | \sigma ^ {r} | $, satisfies the inequalities

$$ {| \sigma ^ {r} | } ^ \prime \leq | \sigma ^ {r} | , $$

$$ {| \partial \sigma ^ {r+1} | } ^ \prime \leq | \sigma ^ {r+1} | , $$

$$ {| T _ {v} \sigma ^ {r} - \sigma ^ {r} | } ^ \prime \leq \frac{ {| \sigma ^ {r} | } ^ \prime | v | }{r+1} , $$

where $ T _ {v} \sigma ^ {r} $ is the cell obtained by shifting by a vector $ v $ of length $ | v | $.

If $ A = \sum a _ {i} \sigma _ {i} ^ {r} $, the sharp norm $ | A | ^ \sharp $ is expressed as follows:

$$ | A | ^ \sharp = \inf \left \{ \frac{\sum | a _ {i} | | \sigma _ {i} ^ {r} | | v _ {i} | }{r+1} + \left | \sum a _ {i} T _ {v _ {i} } \sigma _ {i} ^ {r} \right | ^ \flat \right \} , $$

where $ | C | ^ \flat $ is the flat norm of the chain $ C $, and the infimum is taken over all shifts $ v $.

One has

$$ | a A | ^ \sharp = | a | | A | ^ \sharp , $$

$$ | A + B | ^ \sharp \leq | A | ^ \sharp + | B | ^ \sharp , $$

$$ | A | ^ \sharp = 0 \iff A = 0 , $$

$$ | A | ^ \sharp \leq | A | ^ \flat ; $$

if $ r = 0 $, then $ | A | ^ \sharp = | A | ^ \flat $.

The completion of the space $ C _ {r} ( E ^ {n} ) $ is the separable Banach space $ C _ {r} ^ \sharp ( E ^ {n} ) $, whose elements are known as $ r $-dimensional sharp chains. For any $ r $-dimensional polyhedral chain $ A $ and any vector $ v $,

$$ | T _ {v} A - A | ^ \sharp \leq \frac{| A | | v | }{r+1} , $$

where $ T _ {v} A $ is the chain obtained by shifting $ A $ by the vector $ v $ of length $ | v | $. A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If $ A = \lim\limits ^ \flat A _ {i} $, where $ A _ {i} $ are polyhedral chains, and $ \psi A = \lim\limits ^ \sharp A _ {i} $, where $ \psi $ is a linear bijective mapping from the space $ C _ {r} ^ \flat ( E ^ {n)} $ into the space $ C _ {r} ^ \sharp ( E ^ {n)} $, and $ \psi C _ {r} ^ \flat $ is dense in $ C _ {r} ^ \sharp $ in the sharp norm.

It is not possible to give a correct definition of the boundary $ \partial A $ of a sharp chain [1]; an $ r $- dimensional sharp chain $ X = XA $ is an element of the space $ C ^ {\sharp r } ( E ^ {n} ) $ dual to $ C _ {r} ^ \sharp ( E ^ {n} ) $; it is a flat cochain, and

$$ | X | \leq | X | ^ \flat \leq | X | ^ \sharp , $$

where $ | X | $ is the co-mass of $ X $, while the sharp co-norm $ | X | ^ \sharp $ is defined similarly to the flat norm $ | X | ^ \flat $. The co-boundary $ dX $ of a sharp cochain is not necessarily sharp [1], but

$$ | dX | \leq | X | ^ \flat \leq | X | ^ \sharp . $$

The Lipschitz constant $ {\mathcal L} ( X) $ of a cochain $ X $ is defined as follows:

$$ {\mathcal L} ( X) = \sup \left \{ \frac{| X \cdot ( T _ {v} A - A ) | }{| A | | v | } \right \} , $$

where the $ A $ are polyhedral chains. For sharp cochains this supremum is finite, and

$$ ( r + 1 ) {\mathcal L} ( X) \leq | X | ^ \sharp . $$

Any flat cochain with a finite Lipschitz constant is sharp, and

$$ | X | ^ \sharp = \sup \{ | X | ^ \flat ,\ ( r + 1 ) {\mathcal L} ( X) \} , $$

and also

$$ | dX | \leq ( r + 1 ) {\mathcal L} ( X) . $$

Similar concepts are introduced for $ r $- dimensional polyhedral chains in open subsets $ R \subset E ^ {n} $. See also Sharp form.

The sharp norm in the space of additive functions $ \gamma $ whose values are $ r $- vectors is the largest of the semi-norms $ | \cdot | ^ \prime $ which satisfy the conditions:

$ | \gamma | ^ \prime \leq | \gamma | $, where $ | \gamma | $ is the complete variation of $ \gamma $;

$$ | T _ {v} \gamma - \gamma | ^ \prime \leq \ \frac{| v | | \gamma | }{r+1} , $$

where $ T _ {v} \gamma ( Q) = \gamma T _ {-v} ( Q) $ is the shift of the function $ \gamma $ by the vector $ v $ of length $ | v | $:

$$ T _ {-v} ( Q) = \{ {q - v } : {q \in Q \subset E ^ {n} } \} ; $$

for each point $ p $ and an arbitrary $ \epsilon $ there exists an $ \eta > 0 $ such that $ | \gamma | ^ \prime \leq \epsilon | \gamma | $ if the support $ \supp t \gamma \subset U _ \eta ( p) $ and $ \gamma ( E ^ {n}) = 0 $.

The sharp norm $ | \gamma | ^ \sharp $ is represented as follows:

$$ | \gamma | ^ \sharp = \sup _ \omega \int\limits _ {E ^ {n} } \omega d \gamma , $$

where the $ \omega $ are $ r $- dimensional sharp forms for which $ | \omega | ^ \sharp \leq 1 $.

References

[1] H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957)
How to Cite This Entry:
Sharp norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sharp_norm&oldid=51635
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article