Name of theorem that proves if an integral of a function againist all test functions in a test space is zero, then the function itself must be zero.

Question

I'm curious the name of a theorem in which if: $<f,\phi>_{L_2(\Omega)} = \int_{\Omega} f\phi d\Omega = 0 \; \forall\phi \in \Phi$, then $f=0$.

It seems to me this would be a major result of functional analysis in particular.