  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 $\hat{\omega}(\xi) \leq \frac{\sqrt{5} -
1}{2}$. Besides that, some statements about the dimension of subspaces
generated by best approximations to completely irrational subspace easily
follow from properties that we discuss.
