On the linear independence of a function and its derivatives

We obtain the following results. 1) Suppose that
z(t)
and its first
m
derivatives
z^{(k)}(t), k = 1,...,m
, are continuous functions with values in a normed linear vector space. We define a class of linear functionals and show that if a functional in the class is applied to
z^{(k)}
and vanishes for
0 \leq k \leq m - 1
but does not vanish for
k = m
, then the vectors
\{z^{(k)}(t)\}
are linearly independent for each
t
in the domain of
z(\cdotp)
. 2) If now
z^{(k)}(t), k = 0,...,m
are mean-square continuous random processes such that
z^{(m+1)}(\cdotp)
has a nonvanishing white-noise component, then the random variables
\{z^{(k)}(t)\}, k = 0,..,m
, are linearly independent. These results are shown to be related both in formulation and method of solution.