TY - JOUR

T1 - Is the Free Locally Convex Space L(X) Nuclear?

AU - Leiderman, Arkady

AU - Uspenskij, Vladimir

N1 - Funding Information:
We would like to express our gratitude to Vladimir Pestov and Michael Megrelishvili for the most stimulating questions and discussions on the topic. We thank Mikhail Ostrovskii for guiding us towards the paper [20]. The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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 Außenhofer (Topol Appl 134:90–102, 2007), 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 (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every 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. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set X(1) of all non-isolated points of X is σ-compact.

AB - 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 Außenhofer (Topol Appl 134:90–102, 2007), 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 (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every 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. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set X(1) of all non-isolated points of X is σ-compact.

KW - Free locally convex space

KW - Hilbert space

KW - nuclear locally convex space

KW - reflexive Banach space

KW - Schwartz locally convex space

UR - http://www.scopus.com/inward/record.url?scp=85139801615&partnerID=8YFLogxK

U2 - 10.1007/s00009-022-02178-0

DO - 10.1007/s00009-022-02178-0

M3 - Article

AN - SCOPUS:85139801615

SN - 1660-5446

VL - 19

JO - Mediterranean Journal of Mathematics

JF - Mediterranean Journal of Mathematics

IS - 6

M1 - 241

ER -