Given a real inner product space with the standard Euclidean inner product, I read that equality in the Cauchy-Schwarz Inequality implies linear dependence. Does strict inequality, $|\left<u,v\right>|<||u||*||v||$, imply linear independence?
Actually, $| \langle u,v \rangle | = \|u\| \|v\|$ if and only if $u=\lambda v$ for some scalar $\lambda$. You read the "hard" part of the "iff", since equality is trivial when $u$ and $v$ are linearly dependent.
Hence a strict inequality is a necessary and sufficient condition for the linear independence of $u$ and $v$.