## Abstract

Given a class P of Banach spaces, a locally convex space (LCS) E is called multi-P if E can be isomorphically embedded into a product of spaces that belong to P. We investigate the question whether the free locally convex space L(X) is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive.

If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of [1], L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X.

We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a kω-space, then L(X) is a Schwartz LCS [3], hence L(X) is multi-reflexive. We show that for any paracompact first countable (in particular, metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a kω-space, equivalently, if X is a locally compact and σ-compact space.

If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of [1], L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X.

We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a kω-space, then L(X) is a Schwartz LCS [3], hence L(X) is multi-reflexive. We show that for any paracompact first countable (in particular, metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a kω-space, equivalently, if X is a locally compact and σ-compact space.

Original language | English GB |
---|---|

State | Published - 25 Jun 2021 |

## Keywords

- math.GN
- math.FA
- Primary 46A03, Secondary 46B25, 54D30