  Let $\mathcal{H}_i$ be a finite dimensional complex Hilbert space of
dimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i,
1,2, ..., k$. A subspace $S \subset \mathcal{H} = \mathcal{H}_{A_{1}
  A_{2}... A_{k}} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes ... \otimes
\mathcal{H}_k $ is said to be {\it completely entangled} if it has no nonzero
product vector of the form $u_1 \otimes u_2 \otimes ... \otimes u_k$ with $u_i$
in $\mathcal{H}_i$ for each $i$. Using the methods of elementary linear algebra
and the intersection theorem for projective varieties in basic algebraic
geometry we prove that $$\max_{S \in \mathcal{E}} \dim S = d_1 d_2... d_k -
(d_1 + ... + d_k) + k - 1$$ where $\mathcal{E} $ is the collection of all
completely entangled subspaces.
  When $\mathcal{H}_1 = \mathcal{H}_2 $ and $k = 2$ an explicit orthonormal
basis of a maximal completely entangled subspace of $\mathcal{H}_1 \otimes
\mathcal{H}_2$ is given.
  We also introduce a more delicate notion of a {\it perfectly entangled}
subspace for a multipartite quantum system, construct an example using the
theory of stabilizer quantum codes and pose a problem.
