Chebyshev system

Remarks

Equivalently, the determinant $\det \big( u_i(x_j) \big)_{i, j = 1}^{n}$ is nonzero for all distinct $x_1, \dots, x_n \in K$. Equivalently, for all distinct $x_1, \dots, x_n \in K$ and all $y_1, \dots, y_n \in \R$ there is exactly one combination $u = \sum c_i u_i$ with $u(x_j) = y_j$ for all $j$.

Example: $1, x, \dots, x^{n-1}$ on $[a, b]$. A nonzero polynomial of degree at most $n - 1$ has at most $n - 1$ roots.

Non-example: $1, x^2$ on $[-1, 1]$, with $n = 2$. The combination $x^2 - 1$ is nonzero but has two zeros, $\pm 1$, more than $n - 1 = 1$.

The trigonometric polynomials of degree at most $m$ form a Haar subspace of dimension $2m + 1$ on the circle $\R / 2\pi\Z$: a nonzero one has at most $2m$ zeros in a period.