Computable Versions of the Uniform Boundedness Theorem

We investigate the computable content of the Uniform Boundedness Theorem and of the closely related Banach-Steinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the sequence is “given”. If it is just given with respect to the compact open topology (i.e. if just a sequence of “programs” is given), then we cannot even compute an upper bound of the uniform bound in general. If, however, the pointwise bounds are available as additional input information, then we can effectively compute an upper bound of the uniform bound. Additionally, we prove an effective version of the contraposition of the Uniform Boundedness Theorem: given a sequence of linear bounded operators which is not uniformly bounded, we can effectively find a witness for the fact that the sequence is not pointwise bounded. As an easy application of this theorem we obtain a computable function whose Fourier series does not converge. §