1. The problem statement, all variables and given/known data Suppose u,v ∈ V and that Φ(u)=0 implies Φ(v)=0 for all Φ ∈ V* (the duel space). Show that v=ku for some scalar k. 2. Relevant equations N/A 3. The attempt at a solution I've managed to solve the problem when V is of finite dimension by assuming u,v are linearly independent, expanding them to a basis of V and defining a function such that Φ(v)=0, Φ(u)=1 and Φ(wi)=0 where wi are all the other basis vectors. This contradicted the original statement in the problem (since Φ(v)=0 but Φ(u)=1) which meant u,v must be linearly dependent, which implies u is a scalar multiple of v. However I don't know how to prove it in the general case where V could also be of infinite dimension, the intuitive solution I had was an argument of linear dependence since we have a scalar multiple but I cannot seem to use it in the general case, so this is where I got stuck. Any help would be greatly appreciated.