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The set of all bounded sequences forms the sequence space. [ citation needed ] The definition of boundedness can be generalized to functions f : X → Y {\displaystyle f:X\rightarrow Y} taking values in a more general space Y {\displaystyle Y} by requiring that the image f ( X ) {\displaystyle f(X)} is a bounded set in Y {\displaystyle Y} .
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits , it may be possible to assign a value to a series, called the sum of the series .
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .
Note that this more general concept of boundedness does not correspond to a notion of "size". A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly.
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
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.
Is a subfield of calculus [30] concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. [31] differential equation Is a mathematical equation that relates some function with its derivatives. In applications ...
a sequence = = is said to be Mackey convergent to the origin if there exists a divergent sequence = = of positive real numbers such that the sequence () = is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);