Partitions of unity therefore allow for certain other kinds of function spaces to be considered : for instance L " p " spaces, Sobolev spaces, and other kinds of spaces that require integration.
32.
The Bernstein polynomials of a fixed degree " m " are a family of " m " + 1 linearly independent polynomials that are a partition of unity for the unit interval [ 0, 1 ].
33.
In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of "'R " "'n ".
34.
For instance, the integral of differential forms on paracompact manifolds is first defined locally ( where the manifold looks like Euclidean space and the integral is well known ), and this definition is then extended to the whole space via a partition of unity.
35.
:: In fact, ? can be written as ? = " d " ? with ? a smooth 1-form of compact support : indeed, using partitions of unity, this reduces to the case of a smooth 2-form of compact support on a rectangle.
36.
;Partition of unity : A partition of unity of a space " X " is a set of continuous functions from " X " to [ 0, 1 ] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
37.
;Partition of unity : A partition of unity of a space " X " is a set of continuous functions from " X " to [ 0, 1 ] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
38.
Now, if " f " vanishes on an arbitrary family U _ { \ alpha } of open sets, then for any test function \ phi supported in \ bigcup U _ { \ alpha }, a simple argument based on the compactness of the support of \ phi and a partition of unity shows that f ( \ phi ) = 0 as well.
39.
A variant of ech cohomology, called "'numerable ech cohomology "', is defined as above, except that all open covers considered are required to be " numerable " : that is, there is a partition of unity { ? " i " } such that each support \ { x | \ rho _ i ( x ) > 0 \ } is contained in some element of the cover.
