On Uniform Distribution Modulo 1 and Functional Convergence

In this note, we study the convergence of functional sequences. A criterion for uniform distribution mod 1 is derived. Then we study the partitions, block sequences and the uniform distribution preserving mappings. In the last part, we prove that to each one to one sequence dense in [0, 1] a regular matrix summation method such that this sequence is uniformly distributed mod 1 with respect to this method exists.