As picture below , the $f=0$ means $||f||_{L^2}=0$ , but I don't know how to prove the theory below
2026-04-06 01:51:49.1775440309
How to show $\int f\varphi=0 ~~\Rightarrow ~~f=0$?
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I assume that $\Omega\subset\mathbb{R}^n$ because otherwise it wouldn't make sense to talk about $\varphi$ being $C^\infty$. By the Riesz representation theorem, the dual of $C_0^\infty(\Omega)$ is the space of finite Borel measures on $\Omega$. Then the functional $L:\varphi\mapsto\int_\Omega f(x)\varphi(x)\,dx$ is the zero functional, which means that $\mu=f(x)\,dx$ is the zero measure so $f=0$.