1

I have been struggling a lot proving this theorem, associated to the Fundamental Lemma of CoV.

Let $f,g\in L^1(\Omega)$, where $\Omega\subseteq\mathbb{R}^n$ open. Suppose $\int_{\Omega} f(x)\phi(x) dx=0$ for all test functions $\phi \in C_c^{\infty}(\Omega)$ that satisfy $\int_{\Omega} g(x)\phi(x) dx=0$. Show that there exists $a\in\mathbb{R}$, such that $f = ag$ almost everywhere.

Do you have any hint or solution that I can work with because all my attempts turned out to not solve this task.

Mathemann
  • 59
  • 1
    By linear algebra, there is $a \in \mathbb{R}$ such that $\int f\phi = a\int g\phi$ for all $\phi \in C_c^{\infty}$. – Kakashi Oct 30 '23 at 21:17
  • That would instantly solve it. However I dont know What Theorem you are referring to exactly? – Mathemann Oct 30 '23 at 21:43
  • If $F,G$ are linear mappings from a real vector space $V$ to $\mathbb{R}$ and $\ker G\subset \ker F,$ then $F=aG$ for a real constant $a.$ – Ryszard Szwarc Oct 30 '23 at 22:16
  • Thank you so much for your Help! – Mathemann Oct 31 '23 at 09:54
  • @Mathemann See the answer here: https://math.stackexchange.com/questions/3050228/if-ker-f-subset-ker-g-where-f-g-are-non-zero-linear-functionals-then-show – Kakashi Oct 31 '23 at 13:35

0 Answers0