  Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, and $\mathcal{F}$ be a
field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is
called a product vector if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some
$u^{[1]}\in\mathcal{F}^{d_1},\ldots,u^{[n]}\in\mathcal{F}^{d_n}$. A basis
composed of product vectors is called a product basis. In this paper, we show
that the maximum dimension of subspaces of
$\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ with no product basis
is equal to $d_1d_2\cdots d_n-2$ if either (i) $n=2$ or (ii) $n\ge3$ and
$\#\mathcal{F}>\max\{d_i : i\not=n_1,n_2\}$ for some $n_1$ and $n_2$. When
$\mathcal{F}=\mathbb{C}$, this result is related to the maximum number of
simultaneously distinguishable states in general probabilistic theories (GPTs).
