From Chapter 1, Exercise 14, Strichartz's book: Distribution theory & fourier transforms
- Suppose $f$ & $g$ are distributions such that $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi \rangle = 0$. Show that $\langle f,\phi \rangle =c\langle g,\phi\rangle$ for some constant c.
Does the question mean that if the condition is true for some test functions $\phi$ the result $\langle f,\phi \rangle=c \langle g,\phi\rangle$ can be deduced for all such test functions $\phi$? I think that one has to exploit the continuity of distributions like one has for functions wherein continuous functions are completely defined if they are defined on a dense subset. But apart from that I don't know how to proceed.