On Free Locally Convex Spaces

Let L(X) be the free locally convex space over a Tychonoff space X. We prove that the following assertions are equivalent: (i) every functionally bounded subset of X is finite, (ii) L(X) is semi-reflexive, (iii) L(X) has the Grothendieck property, (iv) L(X) is semi-Montel. We characterize those spaces X, for which L(X) is c0-quasibarrelled, distinguished or a (d f)-space. If X is a convergent sequence, then L(X) has the Glicksberg property, but the space L(X) endowed with its Mackey topology does not have the Schur property.