<P> In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces . </P> <P> An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator . </P> <P> A continuous linear operator maps bounded sets into bounded sets . A linear functional is continuous if and only if its kernel is closed . Every linear function on a finite - dimensional space is continuous . </P> <P> The following are equivalent: given a linear operator A between topological spaces X and Y: </P>

Every linear function on a finite-dimensional space is continuous