Define a map $K:X_n \to X_n$ where $X=\text{span}(v_1, ..., v_n)$ where $v_i$ are basis functions some Hilbert space $H$. So $X_n$ is finite-dimensional.
$B_r(0)$ denotes the ball of radius $r$ centred at $0$ in $X_n$.
(after some calculations)... Therefore, $K$ is locally Lipschitzian and satisfies $K(\overline{B_R(0)}) \subset \overline{B_R(0)}$, thus Brouwer's fixed point theorem tells us there is a fixed point.
Now, why does Brouwer apply here? For Banach spaces doesn't one require compactness of $K(\overline{B_R(0)})$ in $X_n$? Or do we not need it cause of the finite-dimensional nature of $X_n$?
$\overline{B_R(0)}$ is compact, since $X_n$ is finite-dimensional. Since $K$ is locally Lipschitz, it is continuous, and hence $K(\overline{B_R(0)})$ is compact too. But for Brouwer's theorem, we only need the continuity, and that $K(\overline{B_R(0)}) \subset \overline{B_R(0)}$.