  This review paper presents the results, which cover the study of current
problems of approximation theory in abstract linear spaces. Such research has
been actively developed since the 2000s, based on the ideas and approaches
initiated in the articles by Stepanets. In particular, the review contains
results concerning the best, best $n$-term approximations and widths of some
functional compacts in the spaces ${\mathcal S}^p$. Direct and inverse
approximation theorems are also formulated in these spaces.
