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Difference between revisions of "Flat norm"

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in the space  $  E  ^ {n} $''
 
in the space  $  E  ^ {n} $''
  
The norm  $  | A |  ^  \flt $
+
The norm  $  | A |  ^  \flat $
 
which is defined as follows:
 
which is defined as follows:
  
 
$$  
 
$$  
| A |  ^  \flt   = \  
+
| A |  ^  \flat   = \  
 
\inf  \{ | A - \partial  D | + | D | \} ,
 
\inf  \{ | A - \partial  D | + | D | \} ,
 
$$
 
$$
Line 30: Line 30:
  
 
$$  
 
$$  
| a A |  ^  \flt   =  | a | | A |  ^  \flt ,\ \  
+
| a A |  ^  \flat   =  | a | | A |  ^  \flat ,\ \  
| A + B |  ^  \flt   \leq  | A |  ^  \flt + | B |  ^  \flt ,
+
| A + B |  ^  \flat   \leq  | A |  ^  \flat + | B |  ^  \flat ,
 
$$
 
$$
  
 
$$  
 
$$  
| A |  ^  \flt = 0  \iff  A = 0,\  | A |  ^  \flt
+
| A |  ^  \flat = 0  \iff  A = 0,\  | A |  ^  \flat
  \leq  | A |,\  | \sigma |  ^  \flt   =  | \sigma |
+
  \leq  | A |,\  | \sigma |  ^  \flat   =  | \sigma |
 
$$
 
$$
  
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if  $  \pi $
 
if  $  \pi $
 
is the projection of  $  E  ^ {n} $
 
is the projection of  $  E  ^ {n} $
on some plane, one has  $  | \pi A |  ^  \flt \leq  | A |  ^  \flt $.
+
on some plane, one has  $  | \pi A |  ^  \flat \leq  | A |  ^  \flat $.
  
The [[Completion|completion]] of the linear space of polyhedral chains  $  C _ {r} ( E  ^ {n} ) $
+
The [[completion]] of the linear space of polyhedral chains  $  C _ {r} ( E  ^ {n} ) $
is a separable Banach space,  $  C _ {r}  ^  \flt ( E  ^ {n} ) $;  
+
is a separable Banach space,  $  C _ {r}  ^  \flat ( E  ^ {n} ) $;  
 
its elements are known as  $  r $-
 
its elements are known as  $  r $-
dimensional flat chains, to each of which an infinite or a finite mass can be assigned:  $  | A | _  \flt = \inf  \{ {\lim\limits  \inf } _ {i \rightarrow \infty }  | A _ {i} |,  A _ {i} \rightarrow _  \flt A  \textrm{ as  polyhedral  chains  } \} $.
+
dimensional flat chains, to each of which an infinite or a finite mass can be assigned:  $  | A | _  \flat = \inf  \{ {\lim\limits  \inf } _ {i \rightarrow \infty }  | A _ {i} |,  A _ {i} \rightarrow _  \flat A  \textrm{ as  polyhedral  chains  } \} $.
  
 
The boundary  $  \partial  $
 
The boundary  $  \partial  $
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$$  
 
$$  
| \partial  A |  ^  \flt   \leq  | A |  ^  \flt ,\ \  
+
| \partial  A |  ^  \flat   \leq  | A |  ^  \flat ,\ \  
| A |  ^  \flt   =  \inf  \{
+
| A |  ^  \flat   =  \inf  \{
| A - \partial  D |  ^  \flt + | D |  ^  \flt \} .
+
| A - \partial  D |  ^  \flat + | D |  ^  \flat \} .
 
$$
 
$$
  
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$$  
 
$$  
| X \cdot A |  \leq  N | A |  ^  \flt ,
+
| X \cdot A |  \leq  N | A |  ^  \flat ,
 
$$
 
$$
  
 
where  $  | X | $
 
where  $  | X | $
 
is the co-mass of  $  X $.  
 
is the co-mass of  $  X $.  
It is an element of the non-separable space  $  C ^ {\flt r } ( E  ^ {n} ) $
+
It is an element of the non-separable space  $  C ^ {\flat r } ( E  ^ {n} ) $
dual to  $  C _ {r}  ^  \flt ( E  ^ {n} ) $.  
+
dual to  $  C _ {r}  ^  \flat ( E  ^ {n} ) $.  
The flat norm  $  | X |  ^  \flt $
+
The flat norm  $  | X |  ^  \flat $
 
of a flat cochain  $  X $
 
of a flat cochain  $  X $
 
is defined in the standard manner:
 
is defined in the standard manner:
  
 
$$  
 
$$  
| X |  ^  \flt   = \  
+
| X |  ^  \flat   = \  
\sup _ {| A|  ^  \flt = 1 }  | X \cdot A |,
+
\sup _ {| A|  ^  \flat = 1 }  | X \cdot A |,
 
$$
 
$$
  
Line 90: Line 90:
  
 
$$  
 
$$  
| A |  ^  \flt   = \  
+
| A |  ^  \flat   = \  
\sup _ {| X|  ^  \flt = 1 }  | X \cdot A |,\ \  
+
\sup _ {| X|  ^  \flat = 1 }  | X \cdot A |,\ \  
| X \cdot A |  \leq  | X |  ^  \flt | A |  ^  \flt ,
+
| X \cdot A |  \leq  | X |  ^  \flat | A |  ^  \flat ,
 
$$
 
$$
  
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$$  
 
$$  
| X |  \leq  | X |  ^  \flt .
+
| X |  \leq  | X |  ^  \flat .
 
$$
 
$$
  
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$$  
 
$$  
| dX |  ^  \flt   \leq  | X |  ^  \flt ,
+
| dX |  ^  \flat   \leq  | X |  ^  \flat ,
 
$$
 
$$
  
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$$  
 
$$  
| X |  ^  \flt   = \  
+
| X |  ^  \flat   = \  
 
\sup  \{ | X |, | dX | \} .
 
\sup  \{ | X |, | dX | \} .
 
$$
 
$$
  
Similar concepts are introduced for polyhedral  $  r $-
+
Similar concepts are introduced for polyhedral  $  r $-dimensional
dimensional chains situated in open subsets  $  R \subset  E  ^ {n} $.  
+
chains situated in open subsets  $  R \subset  E  ^ {n} $.  
See also [[Flat form|Flat form]].
+
See also [[Flat form]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)  {{MR|0087148}} {{ZBL|0083.28204}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)  {{MR|0087148}} {{ZBL|0083.28204}} </TD></TR></table>

Latest revision as of 19:47, 20 February 2021


of an $ r $- dimensional polyhedral chain $ A $ in the space $ E ^ {n} $

The norm $ | A | ^ \flat $ which is defined as follows:

$$ | A | ^ \flat = \ \inf \{ | A - \partial D | + | D | \} , $$

where $ | C | $ is the mass of the chain $ C $( cf. Mass and co-mass), $ \partial C $ is its boundary, and the infimum is taken over all $ ( r+ 1) $- dimensional polyhedral chains. The properties of the flat norm are

$$ | a A | ^ \flat = | a | | A | ^ \flat ,\ \ | A + B | ^ \flat \leq | A | ^ \flat + | B | ^ \flat , $$

$$ | A | ^ \flat = 0 \iff A = 0,\ | A | ^ \flat \leq | A |,\ | \sigma | ^ \flat = | \sigma | $$

for any cell $ \sigma $; if $ \pi $ is the projection of $ E ^ {n} $ on some plane, one has $ | \pi A | ^ \flat \leq | A | ^ \flat $.

The completion of the linear space of polyhedral chains $ C _ {r} ( E ^ {n} ) $ is a separable Banach space, $ C _ {r} ^ \flat ( E ^ {n} ) $; its elements are known as $ r $- dimensional flat chains, to each of which an infinite or a finite mass can be assigned: $ | A | _ \flat = \inf \{ {\lim\limits \inf } _ {i \rightarrow \infty } | A _ {i} |, A _ {i} \rightarrow _ \flat A \textrm{ as polyhedral chains } \} $.

The boundary $ \partial $ of a flat chain is also defined by passing to the limit; it is a continuous operation, and

$$ | \partial A | ^ \flat \leq | A | ^ \flat ,\ \ | A | ^ \flat = \inf \{ | A - \partial D | ^ \flat + | D | ^ \flat \} . $$

The flat norm is the largest of the semi-norms $ | \cdot | ^ \prime $ on $ C _ {r} ( E ^ {n} ) $ in which all cells $ \sigma $ satisfy the inequalities: $ | \sigma ^ {r} | ^ \prime \leq | \sigma ^ {r} | $, $ | \partial \sigma ^ {r+ 1 } | ^ \prime \leq | \sigma ^ {r+ 1 } | $. An $ r $- dimensional flat cochain $ X $ is a linear function of $ r $- dimensional flat chains $ A $( denoted by $ X \cdot A $) such that, for a given $ N $,

$$ | X \cdot A | \leq N | A | ^ \flat , $$

where $ | X | $ is the co-mass of $ X $. It is an element of the non-separable space $ C ^ {\flat r } ( E ^ {n} ) $ dual to $ C _ {r} ^ \flat ( E ^ {n} ) $. The flat norm $ | X | ^ \flat $ of a flat cochain $ X $ is defined in the standard manner:

$$ | X | ^ \flat = \ \sup _ {| A| ^ \flat = 1 } | X \cdot A |, $$

so that

$$ | A | ^ \flat = \ \sup _ {| X| ^ \flat = 1 } | X \cdot A |,\ \ | X \cdot A | \leq | X | ^ \flat | A | ^ \flat , $$

and

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

For the co-boundary $ dX $ of a flat chain (defined by the condition $ dX \cdot A = X \cdot dA $) one has:

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

so that

$$ | X | ^ \flat = \ \sup \{ | X |, | dX | \} . $$

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

References

[1] H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) MR0087148 Zbl 0083.28204
How to Cite This Entry:
Flat norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flat_norm&oldid=46943
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article