A draft of Bailey's Design of Comparative Experiments may be found here; I am referring to Chapter 2, figure 2.1.
Consider $t$ treatments. We have $N$ plots labeled $1, \dots, N$. We apply treatment $1$ to plots $1, \dots, r_1$; $2$ to plots $r_1 + 1, \dots, r_1 + r_2$, and so forth.
Let $\Omega = \{1, \dots, N\}$ be the set of plots and let $\mathcal{T}$ be the set of treatments, for which $|\mathcal{T}| = t$. We define $T: \Omega \to \mathcal{T}$ as the treatment factor. The treatment subspace $V_T$ consists of the vectors in $\mathbb{R}^{\Omega}$ which are constant on each treatment.
Bailey's text refers to the following figure:
What is the relevance of $V_T$? Usually when running an experiment, I would have the responses $\mathbf{y}$, the treatments $T$, and the covariates (of which there are none here), so I don't quite understand what $V_T$ is. The $\mathbf{u}_i$ seem to be nothing more than vectors of treatment indicators for $A$, $B$, and $C$.
(I believe you can ignore what the "typical $\mathbf{v}$" row is without any issues.)
