A set or sequence of values is called bounded if there is a value \(M\) such that the values are never greater than \(M\) and never smaller than \(-M\). A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element \(x\) the sequence of values of the functions at \(x\) is bounded.


A subset \(E\) of a locally compact topological space is bounded if there exists a compact set \(C\) such that \(E\) is contained in \(C\). Such a subset is called \(s\)-bounded if there exists a sequence of compact sets \(C_i\) such that \(E\) is contained in their union.