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Assume $a<b$. A Chebyshev system of dimension $n+1$ is a family of real continuous functions $u_0,\dots ,u_n$ for which every linear combination $u={\sum }_{j=0}^n{c}_j{u}_j$ with coefficients not all zero has at most $n$ distinct zeros in the interval. In particular the functions are linearly independent: an identically zero nontrivial combination would violate that bound. Multiplicities are not counted in this definition.

For distinct points $x_0,\dots ,x_n$, form the evaluation matrix $A=(u_j(x_i){)}_{i,j=0}^n$. If its determinant vanished, a nonzero vector $c$ would satisfy $Ac=0$. Its associated combination would vanish at all $n+1$ points, contradicting the Chebyshev system property. Conversely, if the property failed, some nonzero coefficient vector would give a combination vanishing at $n+1$ distinct points. At those points $Ac=0$, so the evaluation matrix would be singular. We have proved the determinant criterion for a Chebyshev system:This is also the existence-and-uniqueness condition for interpolation from the span of the functions: any prescribed $n+1$ data values give a uniquely solvable linear system.

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 byFor 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 determinantis 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 every such periodic Chebyshev system has odd dimension; none has even dimension. This determinant argument avoids assuming differentiability or simple zeros of the functions.