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In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ...
Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...
This article is about sets in mathematics. ...
Definition
A set S of real numbers is called bounded above if there is a real number k such that k > s for all s in S. The number k is called an upper bound of S. The terms bounded below and lower bound are similarly defined. A set S is bounded if it is bounded both above and below. Therefore, a set is bounded if it is contained in a finite interval. The text or formatting below is generated by a template which has been proposed for deletion. ...
In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...
Metric space A subset S of a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e. if there exists x in M and r > 0 such that for all s in S, we have d(x, s) < r. M is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Properties which are similar to boundedness but stronger, that is they imply boundedness, are total boundedness and compactness. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...
In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
Relation to boundedness in topological vector spaces In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions are identical but generally this is not the case. In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard...
In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...
In mathematics, given a vector space V over K, a norm on V is a function ||·||:V->R; x->||x|| with the following properties: For all a ∈ K and all u and v ∈ V, 1. ...
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
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