On approximation spaces and Greedy-type bases

The purpose of this paper is to introduce \(\omega \)-Chebyshev–Greedy and \(\omega \)-partially greedy approximation classes and study their relation with \(\omega \)-approximation spaces, where the latter are a generalization of the classical approximation spaces. The relation gives us sufficient conditions of when certain continuous embeddings imply different greedy-type properties. Along the way, we generalize a result by P. Wojtaszczyk as well as characterize semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous result by the first author.