Problems of approximation theory in abstract linear spaces

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.