Generalized functions with pseudobounded support in constructive mathematics

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.