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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.
In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ... converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums ...
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
A real-valued function is bounded if and only if it is bounded from above and below. [ 1 ] [ additional citation(s) needed ] An important special case is a bounded sequence , where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers .
If g is of bounded variation, then it is possible to write = () where g 1 (x) = V x a g is the total variation of g in the interval [a, x], and g 2 (x) = g 1 (x) − g(x). Both g 1 and g 2 are monotone non-decreasing. Now, if f is bounded, the Lebesgue–Stieltjes integral of f with respect to g is defined by
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The first inequality (that is, ‖ ‖ < for all ) states that the functionals in are pointwise bounded while the second states that they are uniformly bounded. The second supremum always equals ‖ ‖ (,) = ‖ ‖, ‖ ‖ and if is not the trivial vector space (or if the supremum is taken over [,] rather than [,]) then closed unit ball can be replaced with the unit sphere
Besides the superposition of different plants growing on the same soil, there is a lateral impact of the higher layers on adjacent plant communities, for example, at the edges of forests and bushes. This particular vegetation structure results in the growth of certain vegetation types such as forest mantle and margin communities. [citation needed]