On the other hand, it is clear that a function can be bounded on each set of an ( infinite ) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a presheaf that in general fails to be a sheaf.
32.
It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves : in all cases the criterion of being a section of the presheaf is " local " in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.
33.
Every abelian von Neumann algebra is isomorphic to a product of ?-finite abelian von Neumann algebras, and every ?-finite abelian von Neumann algebra is isomorphic to a spatial tensor product of discrete abelian von Neumann algebras, i . e ., algebras of bounded functions on a discrete set.
34.
A bounded linear operator is generally not a bounded function; the latter would require that the norm of " L " ( " v " ) be bounded for all " v ", which is not possible unless " Y " is the zero vector space.
35.
In the first extreme case when k = 0 we have a simple circle of radius R, corresponding to the case where C _ i has been shrunk into a point . ( Division by k = 0 in the formula is not a problem since both \ sin and \ cos are bounded functions ).
36.
The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined analogously with respect to any non-negative bounded function \ textstyle v on \ textstyle \ textbf { R } ^ d such that \ textstyle 0 \ leq v ( x ) \ leq 1.
37.
Namely, a polynomial-bounded function is lower elementary if and only if it can be expressed using a composition of the following functions : projections, n + 1, nm, n \, \ stackrel { . } {-} \, m, n \ wedge m, \ lfloor n / m \ rfloor, one exponential function ( 2 ^ n or n ^ m ) with the following restriction on the structure of formulas : the formula can have no more than two floors with respect to an exponent ( for example, xy ( z + 1 ) has 1 floor, ( x + y ) ^ { yz + x } + z ^ { x + 1 } has 2 floors, 2 ^ { 2 ^ x } has 3 floors ).
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