The abbreviation LCS means locally convex space. Let $X$ be a Tychonoff space. The \emph{free locally convex space} on $X$ is a LCS $L(X)$ for which $X$ forms a Hamel basis and such that every continuous mapping $f$ from $X$ to a LCS $E$ lifts to a unique continuous linear operator $\bar f: L(X)\to E$.

\noindent\textbf{Definition} [Multi-$\mathcal{P}$ locally convex space]
Given a class $\sP$ of Banach spaces, an LCS $E$ is called \emph{multi-$\mathcal{P}$} if $E$ can be isomorphically embedded in a product of spaces in $\mathcal{P}$.

\noindent\textbf{Theorem 1.} [Motivating result due to V. Uspenskij (2008)]
For every Tychonoff space $X$ the free LCS $L(X)$ can be isomorphically embedded in the product of Banach spaces of the form $\ell^1(\Gamma)$, in other words, $L(X)$ is multi-$\mathcal{L}^1$, where $\mathcal{L}^1$ is the class of all Banach spaces of the form $\ell^1(\Gamma)$.

\noindent\textbf{Theorem 2.} [Leiderman and Uspenskij, 2022]
(a) If $X$ contains an infinite compact subset then $L(X)$ is not multi-Hilbert.\\
(b) For every Lindel\"of $P$-space $X$, $L(X)$ is multi-Hilbert.

\noindent\textbf{Theorem 3.} [Leiderman and Uspenskij, 2022]
(a) Let $X$ be a locally compact $\sigma$-compact space. Then $L(X)$ is multi-reflexive.\\
(b) Let $X$ be a metrizable space. Then $L(X)$ is multi-reflexive if and only if $X$ is locally compact and has a countable base.

In [Leiderman and Kąkol] we showed that any Banach linear subspace of $L(X)$ is finite-dimensional. As an application of Theorems 1, 3 we will prove the following new

\noindent\textbf{Theorem 4.} For every Tychonoff space $X$ any normed linear subspace of $L(X)$ is finite-dimensional.

Quite recently, jointly with E. Reznichenko and O. Sipacheva we completely resolved a general problem: what are metrizable additive subgroups of $L(X)$?

\noindent\textbf{References}

\noindent A. Leiderman and V. Uspenskij (2022), \emph{Is the free locally convex space $L(X)$ nuclear?} Mediterr. J. Math. 19:241.

\noindent A. Leiderman, J. Kąkol, \emph{Asplund locally convex spaces and the finest locally convex topology}, Results Math. (to appear).