Mairhuber–Curtis theorem

Remarks

The idea of the proof: suppose $(u_1, \dots, u_n)$ is a Chebyshev system, so $D(x_1, \dots, x_n) = \det \big( u_i(x_j) \big)$ is nonzero for distinct points. Choose distinct $x_1, \dots, x_n$ in a small open ball inside $\Omega$. Because $d \ge 2$, one can move $x_1$ and $x_2$ continuously inside the ball so that they exchange places, without colliding with each other or with the other points. Along this motion $D$ changes continuously from $D(x_1, x_2, \dots)$ to $D(x_2, x_1, \dots) = -D(x_1, x_2, \dots)$, so by the intermediate value theorem it vanishes somewhere, a contradiction.

Example: the polynomials of degree at most $1$ in two variables are spanned by $1, x, y$. They are not a Haar space on the unit square: the nonzero function $x - y$ vanishes on the whole diagonal.

In dimension $d = 1$ the argument fails, because two points of an interval cannot exchange places without meeting. Indeed the polynomials of degree at most $n - 1$ form a Haar subspace there.

Going deeper

For a student

The theorem is why multivariate interpolation and uniform approximation behave so differently from the one-variable theory. Uniqueness of best uniform approximation and unisolvence for all point sets cannot hold for a fixed space of functions when $d \ge 2$. Practical methods respond by letting the space depend on the data points, as in radial basis function interpolation, where the basis functions are centred at the data points. The theorem holds more generally for compact Hausdorff spaces: a Haar subspace of dimension at least $2$ exists only if the space embeds in a circle.