In this paper, we discuss some properties of completely irrational subspaces.
We prove that there exist completely irrational subspaces that are badly
approximable and, moreover, sets of such subspaces are winning in different
senses. We get some bounds for Diophantine exponents of vectors that lie in
badly approximable subspaces that are completely irrational; in particular,
for any vector \( \xi \) from two-dimensional badly approximable completely
irrational subspace of \( \mathbb{R}^{d} \) one has

\[
\omega(\xi)\leq \frac{d}{2}-1.
\]

Besides that, some statements about the dimension of subspaces generated by
best approximations to completely irrational subspace easily follow from
properties that we discuss.