Pardon my ignorance, but I don't know much about functional analysis and advanced linear algebra. Let $W$ be a closed convex cone in some Hilbert space (it can be finite, if we need to stipulate that for the following to be true), and let $v$ be some vector in the Hilbert space. Then
If the projection of $v$ onto $W^\perp$ (orthogonal complement) is not zero, then there exists a $w \in W$ such that $\langle v,w \rangle = 0$.
Is something like this true?
Is it necessary that $W$ be a closed convex cone, or can we weaken the conditions on $W$?
Let me try to reword my ideas. Suppose that I have some vector $v$, and I do not know whether it is in $W^\perp$ (in fact, it may not even be in $W^\perp$). What do I need to know in order to conclude there exists a $w \in W$ such that $\langle v,w \rangle = 0$. I was thinking that a nonzero projection of $v$ onto $W^\perp$ would suffice, but perhaps not. SO, what would suffice, given that $W$ is a closed convex cone?