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Adjoint norms (cf. [[Norm|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|Flat norm]]) and sharp chains (see [[Sharp norm|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|Flat norm]].

====Comments====
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|exterior product]]  $  V _ {[} r] $
of a [[Vector space|vector space]]  $  V $. 
Here  ""  denotes exterior product and  $  \beta _ {1} \dots \beta _ {r} \in V $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,   "Geometric measure theory" , Springer  (1969)  pp. Sect. 1.8  {{MR|0257325}} {{ZBL|0176.00801}} </TD></TR></table>