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The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary 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. A bounded set is not necessarily a closed set and vice versa.
A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
Totally bounded space. In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient ...
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and ...
Bounded function. A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics, a function defined on some set with real or complex values is called bounded if the set of its values ...
A set (in light blue) and its boundary (in dark blue). In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking ...
A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. [ 7 ] The term locally convex topological vector space is sometimes shortened to locally convex space or LCTVS. A subset in is called. Convex if for all , ∈, and ≤ ≤, + (−) ∈.
Locally bounded function. A real-valued or complex-valued function defined on some topological space is called a locally bounded functional if for any there exists a neighborhood of such that is a bounded set. That is, for some number one has. In other words, for each one can find a constant, depending on which is larger than all the values of ...