ABSTRACT
The practical method of proving the convergence is based on the Cauchy criterion. This method uses the notion of the fundamental sequence. The convergence for the linear normed spaces is the partial case of the metric convergence. The Cauchy criterion is true for the Euclid space, the space of the continuous functions with the norm the maximum of the absolute value, and some others. The metric convergence is the general enough notion. The weak convergence in the functional spaces is not metrisable; that is, it is impossible to describe by any metric. The metric convergence is the partial case of the topological one. The weak convergence of the unitary spaces is the partial case of the topological one too. Convergence in the sense of the norm of the linear normalized space is called strong convergence.
