  A subspace partition $\Pi$ of $V=V(n,q)$ is a collection of subspaces of $V$
such that each 1-dimensional subspace of $V$ is in exactly one subspace of
$\Pi$. The size of $\Pi$ is the number of its subspaces. Let $\sigma_q(n,t)$
denote the minimum size of a subspace partition of $V$ in which the largest
subspace has dimension $t$, and let $\rho_q(n,t)$ denote the maximum size of a
subspace partition of $V$ in which the smallest subspace has dimension $t$. In
this paper, we determine the values of $\sigma_q(n,t)$ and $\rho_q(n,t)$ for
all positive integers $n$ and $t$. Furthermore, we prove that if $n\geq 2t$,
then the minimum size of a maximal partial $t$-spread in $V(n+t-1,q)$ is
$\sigma_q(n,t)$.
