= Solution
Let $d$ be the dimension of a proposed real <Chebyshev system> on the circle, and regard its <functions> as continuous $2\pi$-periodic <functions> on the real line. Choose lifted points $\theta_0<\theta_1<\cdots<\theta_{d-1}<\theta_0+2\pi$ and set $\theta_d=\theta_0+2\pi$. Move the points by
$$
\theta_i(s)=(1-s)\theta_i+s\theta_{i+1},\qquad 0\le s\le1,\quad 0\le i<d.
$$
For every $s$ the points remain distinct on the circle. Indeed, each consecutive gap is a convex combination of positive gaps; the closing gap is also a convex combination of the original closing gap and $\theta_1-\theta_0$. The evaluation <determinant>
$$
D(s)=\det(u_j(\theta_i(s)))_{i,j=0}^{d-1}
$$
is therefore nonzero for every $s$ by the <determinant criterion for a Chebyshev system>. It is continuous, so the <intermediate value theorem> forbids a sign change. At $s=1$, however, periodicity makes its rows the cyclic permutation of the initial rows, giving $D(1)=(-1)^{d-1}D(0)$. For even $d$ this reverses the sign, a contradiction. Hence \b[every such periodic Chebyshev system has odd dimension; none has even dimension]. This <determinant> argument avoids assuming differentiability or simple zeros of the <functions>.
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