Let V be a vector space of all functions $f:[0,1] \to \mathbb{R}$ where the first and second derivatives exist and the second derivative is continuous. Let $f(1)=f(0)=0$.
Define $<f,g> = \int_{0}^{1} f(x)g(x) dx $.
Let $f_{1}, f_{2}\in V$. Let $L(f_{i}) = \lambda_{i} f_{i}$ for some scalar $\lambda_{i}$.
I need to prove if $\lambda_{1} \neq \lambda_{2}$, then $\int_{0}^{1} f_{1}(x)f_{2}(x) dx =0$.
I know I need to suppose $\lambda_{1} \neq \lambda_{2}$, but I'm having difficulties connecting that to $\int_{0}^{1} f_{1}(x)f_{2}(x) dx =0$.
Any help would be appreciated.
We have $\langle f_1,\,Lf_2\rangle=\lambda_2\langle f_1,\,f_2\rangle$. For Hermitian $L$, which I assume you intended, $\langle f_1,\,Lf_2\rangle=\langle Lf_1,\,f_2\rangle=\lambda_1 \langle f_1,\,f_2\rangle$. Equating these, $(\lambda_1 -\lambda_2) \langle f_1,\,f_2\rangle=0$. If one factor isn't $0$, the other is.