  We consider Kolmogorov widths of finite sets of functions. Any orthonormal
system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated
by linear subspaces of dimension essentially smaller than $N$. This is not true
for weaker metrics: it is known that in every $L_p$, $p<2$, the first $N$ Walsh
functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We
give some sufficient conditions for rigidity. We prove that independence of
functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in
$L_0$ -- the metric that corresponds to convergence in measure. In the case of
$L_p$, $1<p<2$, the condition is weaker: any $S_{p'}$-system is $L_p$-rigid.
Also we obtain some positive results, e.g. that first $N$ trigonometric
functions can be approximated by very-low-dimensional spaces in $L_0$, and by
subspaces generated by $o(N)$ harmonics in $L_p$, $p<1$.
