Physlib.Mathematics.VariationalCalculus.Basic

Let $Y$ be a finite-dimensional real inner product space and $V$ be a real inner product space. Let $\mu$ be a measure on $Y$ that is finite on compact sets and assigns strictly positive measure to every non-empty open set. If $f: Y \to V$ is a continuous function such that for every test function $g: Y \to V$ (a smooth function with compact support), the integral of their inner product vanishes, i.e., $$\int_Y \langle f(x), g(x) \rangle \, d\mu(x) = 0,$$ then $f(x) = 0$ for all $x \in Y$.

Let $X$ be a measurable space where every open set is measurable, and let $V$ be a real inner product space. Let $\mu$ be a measure on $X$ that is finite on compact sets and assigns strictly positive measure to every non-empty open set. If $f: X \to V$ is a test function (a smooth function with compact support) such that for every test function $g: X \to V$, the integral of their inner product vanishes: $$\int_X \langle f(x), g(x) \rangle \, d\mu(x) = 0,$$ then $f(x) = 0$ for all $x \in X$.