  The paper studies the free locally convex space $L(X)$ over a Tychonoff space
$X$. Since for infinite $X$ the space $L(X)$ is never metrizable (even not
Fr\'echet-Urysohn), a possible applicable generalized metric property for
$L(X)$ is welcome. We propose a concept (essentially weaker than
first-countability) which is known under the name a $\mathfrak{G}$-base. A
space $X$ has a {\em $\mathfrak{G}$-base} if for every $x\in X$ there is a base
$\{ U_\alpha : \alpha\in\mathbb{N}^\mathbb{N}\}$ of neighborhoods at $x$ such
that $U_\beta \subseteq U_\alpha$ whenever $\alpha\leq\beta$ for all
$\alpha,\beta\in\mathbb{N}^\mathbb{N}$, where
$\alpha=(\alpha(n))_{n\in\mathbb{N}}\leq \beta=(\beta(n))_{n\in\mathbb{N}}$ if
$\alpha(n)\leq\beta(n)$ for all $n\in\mathbb{N}$. We show that if $X$ is an
Ascoli $\sigma$-compact space, then $L(X)$ has a $\mathfrak{G}$-base if and
only if $X$ admits an Ascoli uniformity $\mathcal{U}$ with a
$\mathfrak{G}$-base. We prove that if $X$ is a $\sigma$-compact Ascoli space of
$\mathbb{N}^\mathbb{N}$-uniformly compact type, then $L(X)$ has a
$\mathfrak{G}$-base. As an application we show: (1) if $X$ is a metrizable
space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ is
$\sigma$-compact, and (2) if $X$ is a countable Ascoli space, then $L(X)$ has a
$\mathfrak{G}$-base if and only if $X$ has a $\mathfrak{G}$-base.
