This is a follow-up (of sorts) to this question.

Let $f : X \to T$ be a proper morphism of schemes. Then the notion of a relative ample (or $f$-ample) line bundle can be defined in several equivalent ways; Lazarsfeld's book gives the clearest discussion I have found. The condition that $L$ is $f$-ample is equivalent to each of the following:

$L\big\vert_{f^{-1}(t)}$ is ample on $f^{-1}(t)$ for all $t \in T$.

If $D$ is the divisor class corresponding to $L$, then $D^{\dim V}\cdot V > 0$ for each subvariety of $X$ which maps to a point in $T$.

Given an ample line bundle $A$ on $T$, $L\otimes f^*A^{\otimes m}$ is ample on $X$ for sufficiently large positive $m$.

My question is: are these statements still equivalent if we only know, *a priori*, that $X$ is an algebraic space? I suspect they do, and that the proofs in Lazarsfeld basically go through the same, but an independent reference would be welcome if one exists.

Edit: I'm not sure whether it matters, but I'm only really interested in the case where $X$ is smooth (i.e. a Moishezon manifold).