Bounded linear function - Definition and Overview

In mathematics, a bounded linear function is a linear function which satisfies a particularly property of boundedness. This is usually local boundedness. A function f: X→Y is said to be locally bounded if for each x ∈ X, there exists a neighborhood C of x such that for each y∈C, |f(y)| ≤ L < ∞.

Note that the definition of local boundedness is only meaningful if a topology is specified for X and if there exists a norm for Y, so in the context of vector spaces, both X and Y must be normed vector spaces.

It is useful to consider bounded linear functions as there exist rather esoteric linear functions which are not bounded, and do not have some of the properties we might expect a linear function to have. For example, there exist functions f: R→R which are linear (over Q) but not of the form f(x) = ax.