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''of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406101.png" />-dimensional polyhedral chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406102.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406103.png" />''
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The norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406104.png" /> which is defined as follows:
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{{TEX|done}}
  
<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/f/f040/f040610/f0406105.png" /></td> </tr></table>
+
''of an  $  r $-
 +
dimensional polyhedral chain  $  A $
 +
in the space  $  E  ^ {n} $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406106.png" /> is the mass of the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406107.png" /> (cf. [[Mass and co-mass|Mass and co-mass]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406108.png" /> is its boundary, and the infimum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f0406109.png" />-dimensional polyhedral chains. The properties of the flat norm are
+
The norm  $  | A | ^  \flt  $
 +
which is defined as follows:
  
<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/f/f040/f040610/f04061010.png" /></td> </tr></table>
+
$$
 +
| A |  ^  \flt  = \
 +
\inf  \{ | A - \partial  D | + | D | \} ,
 +
$$
  
<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/f/f040/f040610/f04061011.png" /></td> </tr></table>
+
where  $  | C | $
 +
is the mass of the chain  $  C $(
 +
cf. [[Mass and co-mass|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
  
for any cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061012.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061013.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061014.png" /> on some plane, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061015.png" />.
+
$$
 +
| a A |  ^  \flt  = | a | | A |  ^  \flt  ,\ \
 +
| A + B |  ^  \flt  \leq  | A |  ^  \flt  + | B |  ^  \flt  ,
 +
$$
  
The [[Completion|completion]] of the linear space of polyhedral chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061016.png" /> is a separable Banach space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061017.png" />; its elements are known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061018.png" />-dimensional flat chains, to each of which an infinite or a finite mass can be assigned: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061019.png" />.
+
$$
 +
| A |  ^  \flt  = 0  \iff  A = 0,\  | A |  ^  \flt
 +
\leq  | A |,\  | \sigma |  ^  \flt  = | \sigma |
 +
$$
  
The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061020.png" /> of a flat chain is also defined by passing to the limit; it is a continuous operation, and
+
for any cell  $  \sigma $;
 +
if  $  \pi $
 +
is the projection of  $  E  ^ {n} $
 +
on some plane, one has  $  | \pi A |  ^  \flt  \leq  | A |  ^  \flt  $.
  
<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/f/f040/f040610/f04061021.png" /></td> </tr></table>
+
The [[Completion|completion]] of the linear space of polyhedral chains  $  C _ {r} ( E  ^ {n} ) $
 +
is a separable Banach space,  $  C _ {r}  ^  \flt  ( 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 | _  \flt  = \inf  \{ {\lim\limits  \inf } _ {i \rightarrow \infty }  | A _ {i} |,  A _ {i} \rightarrow _  \flt  A  \textrm{ as  polyhedral  chains  } \} $.
  
The flat norm is the largest of the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061023.png" /> in which all cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061024.png" /> satisfy the inequalities: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061026.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061027.png" />-dimensional flat cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061028.png" /> is a linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061029.png" />-dimensional flat chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061030.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061031.png" />) such that, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061032.png" />,
+
The boundary  $  \partial  $
 +
of a flat chain is also defined by passing to the limit; it is a continuous operation, and
  
<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/f/f040/f040610/f04061033.png" /></td> </tr></table>
+
$$
 +
| \partial  A |  ^  \flt  \leq  | A |  ^  \flt  ,\ \
 +
| A |  ^  \flt  = \inf  \{
 +
| A - \partial  D |  ^  \flt  + | D |  ^  \flt  \} .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061034.png" /> is the co-mass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061035.png" />. It is an element of the non-separable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061036.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061037.png" />. The flat norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061038.png" /> of a flat cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061039.png" /> is defined in the standard manner:
+
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 $,
  
<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/f/f040/f040610/f04061040.png" /></td> </tr></table>
+
$$
 +
| X \cdot A |  \leq  N | A |  ^  \flt  ,
 +
$$
 +
 
 +
where  $  | X | $
 +
is the co-mass of  $  X $.
 +
It is an element of the non-separable space  $  C ^ {\flt r } ( E  ^ {n} ) $
 +
dual to  $  C _ {r}  ^  \flt  ( E  ^ {n} ) $.
 +
The flat norm  $  | X |  ^  \flt  $
 +
of a flat cochain  $  X $
 +
is defined in the standard manner:
 +
 
 +
$$
 +
| X |  ^  \flt  = \
 +
\sup _ {| A|  ^  \flt  = 1 }  | X \cdot A |,
 +
$$
  
 
so that
 
so 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/f/f040/f040610/f04061041.png" /></td> </tr></table>
+
$$
 +
| A |  ^  \flt  = \
 +
\sup _ {| X|  ^  \flt  = 1 }  | X \cdot A |,\ \
 +
| X \cdot A |  \leq  | X |  ^  \flt  | A |  ^  \flt  ,
 +
$$
  
 
and
 
and
  
<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/f/f040/f040610/f04061042.png" /></td> </tr></table>
+
$$
 +
| X |  \leq  | X |  ^  \flt  .
 +
$$
  
For the co-boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061043.png" /> of a flat chain (defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061044.png" />) one has:
+
For the co-boundary $  dX $
 +
of a flat chain (defined by the condition $  dX \cdot A = X \cdot dA $)  
 +
one has:
  
<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/f/f040/f040610/f04061045.png" /></td> </tr></table>
+
$$
 +
| dX |  ^  \flt  \leq  | X |  ^  \flt  ,
 +
$$
  
 
so that
 
so 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/f/f040/f040610/f04061046.png" /></td> </tr></table>
+
$$
 +
| X |  ^  \flt  = \
 +
\sup  \{ | X |, | dX | \} .
 +
$$
  
Similar concepts are introduced for polyhedral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061047.png" />-dimensional chains situated in open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040610/f04061048.png" />. See also [[Flat form|Flat form]].
+
Similar concepts are introduced for polyhedral $  r $-
 +
dimensional chains situated in open subsets $  R \subset  E  ^ {n} $.  
 +
See also [[Flat form|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>

Revision as of 19:39, 5 June 2020


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

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

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

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

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

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

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

The completion of the linear space of polyhedral chains $ C _ {r} ( E ^ {n} ) $ is a separable Banach space, $ C _ {r} ^ \flt ( 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 | _ \flt = \inf \{ {\lim\limits \inf } _ {i \rightarrow \infty } | A _ {i} |, A _ {i} \rightarrow _ \flt 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

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

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 $,

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

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

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

so that

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

and

$$ | X | \leq | X | ^ \flt . $$

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

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

so that

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

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=28192
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article