Continuous piecewise linear functions

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on the m-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean form (Y) over bar (j epsilon J) (boolean OR) (i epsilon Ej) boolean AND (r(i)(0) + r(i)(1) X-1 +r(i)(2) X-2 + center dot center dot center dot + r(i)(m) X-m). where X = (X-1, X-2,..., X-m) is some random vector, {E-j}(j epsilon J) is a finite family of finite sets.