Inner product smallness condition for linear independence of unit vectors. Suppose $m \geq k$ and  $v_1,...,v_k$ are unit vectors living in $\mathbb{R}^m$ (or maybe even an infinite dimensional Hilbert space). 
It is clear that if they are pairwise orthogonal then they are linearly independent. However, I was wondering if there is a constant, $c$ (maybe depending on $m$, $k$) so that $\langle v_i, v_k \rangle < c$ implies that $v_1,...,v_k$ are linearly independent. I suspect there is due to the fact that the inner product is continuous, but I've had some trouble writing down an explicit proof.
 A: Suppose $n \ge 2$ and $v_1, \ldots, v_k \in \mathbb{R}^n$ satisfy $|\langle v_i, v_j \rangle| < \frac{1}{n-1}$ for all $i \neq j$. Then the Gram matrix, formed by $(A)_{i,j} = \langle v_i, v_j \rangle$ is a positive semidefinite matrix that satisfies the conditions for this question, and hence the matrix is invertible. It follows from the properties of the Gram matrix that the list is linearly independent.
In infinite dimensions, there is no corresponding result. To see this, take any infinite orthonormal sequence $(e_n)_{n=1}^\infty$ in the space. Suppose such a $c$ exists as in the question, and choose $n$ such that $\frac{1}{\sqrt{n}} < c$. Then, let
$$f = \frac{e_1 + \ldots + e_n}{\|e_1 + \ldots + e_n\|} = \frac{e_1 + \ldots + e_n}{\sqrt{n}}.$$
Consider the list $e_1, e_2, \ldots, e_n, f$. Note that it's linearly dependent, but we have $\langle e_i, f \rangle = \frac{1}{\sqrt{n}} < c$, and $\langle e_i, e_j\rangle = 0 < c$ for $i \neq j$.
A: The $v_i$ are linearly dependent exactly when one of a certain set of determinants vanishes.  (The set of all the $k\times k$ minors of the matrix formed by the entries in the $v_i$.)  The determinant is a continuous function, so the set of matrices corresponding to independent $(v_1,\dots,v_k)$ is open.  Your all-orthogonal configuration corresponds to all the determinants in question evaluating to $\pm 1$, so there is an open ball's worth of deformations of your $v_i$ on which  independence is preserved.
