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A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
Each set has a supremum (infimum), if it is bounded from above (below). Proof: Without loss of generality one can look at a set A ⊂ R {\displaystyle A\subset \mathbb {R} } that has an upper bound. One can now construct a sequence ( I n ) n ∈ N {\displaystyle (I_{n})_{n\in \mathbb {N} }} of nested intervals I n = [ a n , b n ] {\displaystyle ...
A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R 2 constrained by two parabolic curves x 2 + 1 and x 2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).
An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite ...
Indeed, the elements of define a pointwise bounded family of continuous linear forms on the Banach space := ′, which is the continuous dual space of . By the uniform boundedness principle, the norms of elements of S , {\displaystyle S,} as functionals on X , {\displaystyle X,} that is, norms in the second dual Y ″ , {\displaystyle Y'',} are ...
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X.