# Flat norm

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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=51631
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article