Recently I have learned that on some math department the introductory course to Lebesgue integration is not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is defined as the abstract completion of the space of continuous functions with respect to the norm defined by the Riemann integral of the square of a function.

I did not know that such an approach is possible. This motivated the question in the title. Particularly I would like to address it to algebraists and differential geometers, in other words whether the Riemann integral is sufficient for their professional life.