Albiac Alesanco, Fernando JoséAnsorena, José L.Cúth, MarekDoucha, Michal2020-05-122022-03-0120200022-123610.1016/j.jfa.2019.108354https://academica-e.unavarra.es/handle/2454/36849Our goal in this paper is to continue the study initiated by the authors in of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0 < p ≤ 1, denoted Fp(M). Here we develop new techniques to show that, by analogy with the case p = 1, the space p embeds isomorphically in Fp(M) for 0 < p < 1. Going further we see that despite the fact that, unlike the case p = 1, this embedding need not be complemented in general, complementability of p in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in Fp([0, 1]), we obtain examples of p-Banach spaces for p < 1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis.34 p.application/pdfeng© 2019 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0.Quasimetric spaceQuasi-Banach spaceLipschitz free p-spaceEmbedding of ℓpinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccess