68T05, 30C40, 44A05, 35A22.

In this paper, in connection with a basic formula by S. Smale and D. X. Zhou which is fundamental in the approximation error estimates in statistical learning theory, we give new norm type inequalities based on the general theory of reproducing kernels combined with linear mappings in the framework of Hilbert spaces. We shall give typical concrete examples which may be considered as new norm type inequalities and have physical meanings.;