# Mass and co-mass

Adjoint norms (cf. Norm) in certain vector spaces dual to each other.

1) The mass of an -vector , i.e. an element of the -fold exterior product of a vector space, is the number

The co-mass of an -covector is the number

Here is the standard norm of an -vector and is the scalar product of a vector and a covector.

The mass and the co-mass are adjoint norms in the spaces of -vectors and -covectors , respectively. In this connection:

a) , ;

b) , , and equalities hold if and only if () is a simple -(co)vector;

c) , for exterior products , where for a simple multi-covector (or ) , and, in general, if and ;

d) for inner products , where for and for , and .

These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are and . For example, the co-mass of a form on a domain is

2) The mass of a polyhedral chain is

where is the volume of the cell . 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 is defined in the standard way:

where is a polyhedral chain and is the value of the cochain on the chain .

For references see Flat norm.