Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, locally convex space (LCS) E is called multi- if can be isomorphically embedded into product spaces that belong to . We investigate the question whether free L(X) strongly nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X Tychonoff containing an infinite compact subset then, as it follows from results Außenhofer (Topol Appl 134:90–102, 2007), not nuclear. prove for such LCS has stronger property being multi-Hilbert. deduce k-space, then following properties are equivalent: (1) nuclear; (2) (3) multi-Hilbert; (4) countable and discrete. On other hand, we show nuclear every projectively P-space (in particular, Lindelöf P-space) X. observe Schwartz It known $$k_\omega $$ -space, (Außenhofer et al. in Stud Math 181(3):199–210, hence first-countable paracompact metrizable) converse true, so multi-reflexive only equivalently $$\sigma -compact space. Similarly, any abelian topological group A(X) set $$X^{(1)}$$ all non-isolated points -compact.