The Banach-Steinhaus theorem for (LF)-spaces in constructive analysis

The space D(R) of test functions (infinitely differentiable functions with compact support) is an important example of a non-metrizable (LF)-space, and the domain of distributions (generalized functions). E. Bishop suggested in [1, Appendix A] and [2, Chapter 7, Notes] that the completeness of D(R) and the weak completeness of its dual space would not hold in Bishop’s constructive mathematics. This matter had not be solved since Bishop referred it, and we first obtained the following consequence in [8, Theorem 4]: the completeness of D(R) is equivalent to a principle BD-N, which can be proved in classical mathematics, intuitionistic mathematics of L. E. J. Brouwer and constructive recursive mathematics of A. A. Markov’s school but cannot be in Bishop’s framework (see [4] and [6] for more details). Therefore, in Bishop’s framework, the completeness of a D(R) cannot be proved, and neither can that of a (LF)-space. On the other hand, we can prove the following version of the Banach-Steinhaus theorem for D(R) in Bishop’s constructive mathematics, and therefore can in the others, since theorems in Bishop’s framework belong to the others (see [3, Chapter 1]):