# Mass and co-mass

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.

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 $\beta _ {1} \dots \beta _ {r} \in V$.