The uniform boundedness theorem and a boundedness principle

摘要

We deal with a form of the uniform boundedness theorem (or the Banach-Steinhaus theorem) for topological vector spaces in Bishop's constructive mathematics, and show that the form is equivalent to the boundedness principle BD-N, and hence holds not only in classical mathematics but also in intuitionistic mathematics and in constructive recursive mathematics. The result is also a result in constructive reverse mathematics.