Flat norm

of an -dimensional polyhedral chain in the space

The norm which is defined as follows:

where is the mass of the chain (cf. Mass and co-mass), is its boundary, and the infimum is taken over all -dimensional polyhedral chains. The properties of the flat norm are

for any cell ; if is the projection of on some plane, one has .

The completion of the linear space of polyhedral chains is a separable Banach space, ; its elements are known as -dimensional flat chains, to each of which an infinite or a finite mass can be assigned: .

The boundary of a flat chain is also defined by passing to the limit; it is a continuous operation, and

The flat norm is the largest of the semi-norms on in which all cells satisfy the inequalities: , . An -dimensional flat cochain is a linear function of -dimensional flat chains (denoted by ) such that, for a given ,

where is the co-mass of . It is an element of the non-separable space dual to . The flat norm of a flat cochain is defined in the standard manner:

so that

and

For the co-boundary of a flat chain (defined by the condition ) one has:

so that

Similar concepts are introduced for polyhedral -dimensional chains situated in open subsets . See also Flat form.

References

[1]	H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) MR0087148 Zbl 0083.28204