Let $S={\rm span}(v_1,\ldots,v_k)$ be a subspace of $R^n$, and let $P_S$ be the orthogonal projection matrix onto $S$. If for some $x$ we have $x^T v_i = 0$ for all $i=1,\ldots,k$, then we can conclude that $P_S x=0$.
I would like a quantitative version of this statement which I'll now describe. Suppose that instead we just know that $|x^T v_i| \leq \epsilon$ for all $i=1,\ldots,k$; and moreover $||x||_2=1$, $||v_i||_2=1$ for all $i=1,\ldots,k$. I'd like an argument that concludes $||P_S x||_2 \leq f(\epsilon, v_1, \ldots, v_n)$ where $f(\epsilon, v_1, \ldots, v_n)$ is some function which approaches $0$ as $\epsilon \rightarrow 0$ and the other arguments are kept fixed. Naturally, the faster rate of decay as $\epsilon \rightarrow 0$, the better.