If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.Ī set of real numbers is bounded if and only if it has an upper and lower bound. In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. Main article: Bounded set (topological vector space) This is also called the Heine-Borel theorem.īoundedness in topological vector spaces

A subset of Euclidean space R n is compact if and only if it is closed and bounded.

A metric space is compact if and only if it is complete and totally bounded.

For subsets of R n the two are equivalent.

Total boundedness implies boundedness.

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. Therefore, a set of real numbers is bounded if it is contained in a finite interval.Ī subset S of a metric space ( M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The terms bounded from below and lower bound are similarly defined.Ī set S is bounded if it has both upper and lower bounds. The number k is called an upper bound of S. 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).ĭefinition in the real numbers A real set with upper bounds and its supremum.Ī set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥ s for all s in S. The word 'bounded' makes no sense in a general topological space without a corresponding metric.īoundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.Ī bounded set is not necessarily a closed set and vice versa.

Conversely, a set which is not bounded is called unbounded.

In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. The set at the bottom continues forever towards the right. Often one has a set of nice functions and a way of measuring distances between them.Collection of mathematical objects of finite size An artist's impression of a bounded set (top) and of an unbounded set (bottom). Completion is particularly common as a tool in functional analysis.

For example, in abstract algebra, the p-adic numbers are defined as the completion of the rationals under a different metric. Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, is the completion of (0, 1), and the real numbers are the completion of the rationals. In fact, every metric space has a unique completion, which is a complete space that contains the given space as a dense subset. This notion of 'missing points' can be made precise. In the Euclidean metric, the green path has length 6 2 ≈ 8.49 is complete but the homeomorphic space (0, 1) is not. In the taxicab metric the red, yellow and blue paths have the same length (12), and are all shortest paths. The plane (a set of points) can be equipped with different metrics. Mathematical space with a notion of distance

Unbounded metric space