Prove that :
$$\int_{a}^{b} (\varphi (t)-\lambda_{\varphi}\theta(t))\,dt =0$$
Where :
$$\theta , \varphi \in D(]a,b[) :\text{functions test} $$ Such that :
$$\lambda_{\varphi}=\displaystyle\int_{a}^{b} \varphi (t)dt$$
And :
$$\displaystyle\int_{a}^{b} \theta (t) dt=1$$
- also prove that $$\exists\psi\in D(]a,b[) , \varphi =\psi '+\lambda_{\varphi}\theta $$ My try :
$$\displaystyle\int_{a}^{b} (\varphi (t)-\lambda_{\varphi}\theta(t))dt =\displaystyle\int_{a}^{b} \varphi (t)-\displaystyle\int_{a}^{b}\displaystyle\int_{a}^{b} \varphi (t)\theta (t) dtdt$$ From here how I can complete ?
Can you assist
Thanks!