The Mackey problem for free locally convex spaces

It is known that the free locally convex space {L(X)} on a space X is metrizable only if X is finite and that {L(X)} is barrelled if and only if X is discrete. We significantly generalize these results by proving that {L(X)} is a Mackey space if and only if X is discrete. Noting that real locally convex spaces which are Mackey groups are always Mackey spaces, but that the converse is false, it is also proved here that {L(X)} is a Mackey group if and only if it is a Mackey space.

Free Subspaces of Free Locally Convex Spaces

IfXandYare Tychonoff spaces, letL(X)andL(Y)be the free locally convex space overXandY, respectively. For generalXandY, the question of whetherL(X)can be embedded as a topological vector subspace ofL(Y)is difficult. The best results in the literature are that ifL(X)can be embedded as a topological vector subspace ofL(I), whereI=[0,1], thenXis a countable-dimensional compact metrizable space. Further, ifXis a finite-dimensional compact metrizable space, thenL(X)can be embedded as a topological vector subspace ofL(I). In this paper, it is proved thatL(X)can be embedded inL(R)as a topological vector subspace ifXis a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case ifX=Rn, n∈N.It is also shown that ifGandQdenote the Cantor space and the Hilbert cubeIN, respectively, then (i)L(X)is embedded inL(G)if and only ifXis a zero-dimensional metrizable compact space; (ii)L(X)is embedded inL(Q)if and only ifYis a metrizable compact space.

Free Locally Convex Spaces and the k-space Property

Let L(X) be the free locally convex space over a Tychonoff space X. Then L(X) is a k-space if and only if X is a countable discrete space. We prove also that L(D) has uncountable tightness for every uncountable discrete space D.