Generalized functions with pseudobounded support in constructive mathematics

Abstract

In this paper we define pseudoboundedness for support of a distribution which is weaker than boundedness in Bishop's constructive mathematics. We prove in Bishop's framework that a distribution (sequentially continuous linear functional on the space D ( R ) of test functions) with pseudobounded support is a sequentially continuous linear functional on the space E ( R ) of infinitely differentiable functions on R . We also show that the following three propositions can be proved in classical mathematics, Brouwer's intuitionistic mathematics and constructive recursive mathematics of Markov's school, but cannot be in Bishop's framework: every sequentially continuous linear functional on E ( R ) is bounded on E ( R ) ; every bounded distribution with pseudobounded support is bounded on E ( R ) ; every sequentially continuous linear functional on E ( R ) is a distribution with compact support.