  In this note, we offer a short proof of V. V. Shchigolev's result that over
any field k of characteristic p>2, the T-space generated by
x_1^p,x_1^px_2^p,... is finitely based, which answered a question raised by A.
V. Grishin. More precisely, we prove that for any field of any positive
characteristic, R_2^{(d)}=R_3^{(d)} for every positive integer d, and that over
an infinite field of characteristic p>2, L_2=L_3. Moreover, if the
characteristic of k does not divide d, we prove that R_1^{(d)} is an ideal of
k_0<X> and thus in particular, R_1^{(d)}=R_2^{(d)}. Finally, we show that for
any field of characteristic p>2, R_1^{(d)} is not equal to R_2^{(d)} and L_1 is
not equal to L_2.
