Free locally convex spaces with a small base

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 Fréchet-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 G-base. A space X has a G-base if for every x∈ X there is a base { U_α: α∈ N^N} of neighborhoods at x such that U_β⊆ U_α whenever α≤ β for all α, β∈ N^N, where α= (α(n)) _{n ∈ N}≤ β= (β(n)) _{n ∈ N} if α(n) ≤ β(n) for all n∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of N^N-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.