I got stuck on this problem that I'm working with:
Let $g \in C[0,1]$ s.t for all $f \in C[0,1]$ we have: $$ \left| \int_0^1 f(x^2)g(x)dx \right| \leq \int_0^1 |f(x) |dx $$ Show that $|g(x)| \leq 2x, x \in [0,1]$.
My attempt: Since the functional (call it $T$) is bounded in $L^1$-norm and $C[0,1]$ is dense in $L^1$ the functional can be extended uniquely to all of $L^1$ with the same bound: $ |\widetilde{T}(f)| \leq \lVert f \rVert_1, \forall f \in L^1$. Then by Riesz representation theorem there is a unique function $\psi \in L^{\infty}$ such that $\widetilde{T}(f)=\int_0^1f(x) \psi(x)dx \ \ \forall f \in L^1$.
After this I'm not sure how to proceed (or if this is the right way to go). Do you have any ideas?