Sharp form
An -dimensional differential form
on an open subset
such that the co-mass (cf. Mass and co-mass)
and the co-mass Lipschitz constant
![]() |
where and
is the length of the vector
, are finite. The number
![]() |
is known as the sharp norm of the form .
Whitney's theorem. To each -dimensional sharp cochain
in
corresponds a unique
-dimensional sharp form
for which
![]() |
for all -dimensional oriented simplices
;
is defined by the formula
![]() |
where is a sequence of simplices containing the point
, with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains
into the space
of sharp forms; moreover:
, i.e. the co-mass of
;
, i.e. the Lipschitz constant of
;
, i.e. the sharp norm of
;
is a Banach space.
In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).
The space of
-dimensional sharp chains
of finite mass
with the sharp norm
is isomorphic to the space
of additive set functions whose values are
-vectors
, provided with the sharp norm
; this correspondence is defined by the formula:
![]() | (*) |
for any cochain , where
is the
-dimensional sharp form corresponding to the cochain
, and
, i.e. the covector of the chain
;
, i.e. the complete variation of
;
, i.e. the sharp norm of the chain
.
Thus, (*) is a generalization of the ordinary Lebesgue–Stieltjes integral. In particular, the Lebesgue-measurable summable function associated with
(cf. Flat form), i.e.
![]() |
for any cochain , exists for
if and only if
is absolutely continuous.
If is a regular form and
is a sharp cochain, then there exists a form
, and Stokes' formula
![]() |
applies. Other results established for regular forms can be generalized in an analogous manner.
For references see Sharp norm.
Sharp form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sharp_form&oldid=48680