Properties
If is any set of real numbers then if and only if and otherwise
If are sets of real numbers then (unless ) and
Identifying infima and suprema
If the infimum of exists (that is, is a real number) and if is any real number then if and only if is a lower bound and for every there is an with
Similarly, if is a real number and if is any real number then if and only if is an upper bound and if for every there is an with
Relation to limits of sequences
If is any non-empty set of real numbers then there always exists a non-decreasing sequence in such that Similarly, there will exist a (possibly different) non-increasing sequence in such that
Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from topology that if is a continuous function and is a sequence of points in its domain that converges to a point then necessarily converges to
It implies that if is a real number (where all are in ) and if is a continuous function whose domain contains and then
which (for instance) guarantees[note 1] that is an adherent point of the set
If in addition to what has been assumed, the continuous function is also an increasing or non-decreasing function, then it is even possible to conclude that
This may be applied, for instance, to conclude that whenever is a real (or complex) valued function with domain whose sup norm is finite, then for every non-negative real number
since the map defined by is a continuous non-decreasing function whose domain always contains and
Although this discussion focused on similar conclusions can be reached for with appropriate changes (such as requiring that be non-increasing rather than non-decreasing). Other norms defined in terms of or include the weak space norms (for ), the norm on Lebesgue space and operator norms. Monotone sequences in that converge to (or to ) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.
Arithmetic operations on sets
The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets.
Throughout, are sets of real numbers.
Sum of sets
The Minkowski sum of two sets and of real numbers is the set
consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfies
and
Product of sets
The multiplication of two sets and of real numbers is defined similarly to their Minkowski sum:
If and are nonempty sets of positive real numbers then and similarly for suprema [3]
Scalar product of a set
The product of a real number and a set of real numbers is the set
If then
while if then
Using and the notation it follows that
Multiplicative inverse of a set
For any set that does not contain let
If is non-empty then
where this equation also holds when if the definition is used.[note 2]
This equality may alternatively be written as
Moreover, if and only if where if[note 2] then