  In 2008, L\'aszl\'o Z\'adori proved that the lattice Sub$(V)$ of all
subspaces of a vector space $V$ of finite dimension at least 3 over a finite
field $F$ has a 5-element generating set; in other words, Sub$(V)$ is
5-generated. We prove that the same holds over every 1- or 2-generated field;
in particular, over every field that is a finite degree extension of its prime
field. Furthermore, let $F$, $t$, $V$, $d\geq 3$, $[d/2]$, and $m$ denote an
arbitrary field, the minimum cardinality of a generating set of $F$, a finite
dimensional vector space over $F$, the dimension (assumed to be at least $3$)
of $V$, the integer part of $d/2$, and the least cardinal such that $m[d^2/4]$
is at least $t$, respectively. We prove that Sub$(V)$ is $(4+m)$-generated but
none of its generating sets is of size less than $m$. Moreover, the $k$-th
direct power of Sub$(V)$ is $(5+m)$-generated for many positive integers $k$;
for all positive integers $k$ if $F$ is infinite. Finally, let $n$ be a
positive integer. For $i=1,\dots, n$, let $p_i$ be a prime number or 0, and let
$V_i$ be the 3-dimensional vector space over the prime field of characteristic
$p_i$. We prove that the direct product of the lattices Sub$(V_1)$, ...,
Sub$(V_n)$ is 4-generated if and only if each of the numbers $p_1$, ..., $p_n$
occurs at most four times in the sequence $p_1$, ..., $p_n$. Neither this
direct product nor any of the subspace lattices Sub$(V)$ above is 3-generated.
