Let $a,b,g,h$ be real numbers. How to prove that the functional $F\colon L^2 [a,b]\to \mathbb{R}$, given by $F(u)=\int_a^b (u^2(x)-gu(x)-h)\,dx$ is continuous?
Thank you
Let $a,b,g,h$ be real numbers. How to prove that the functional $F\colon L^2 [a,b]\to \mathbb{R}$, given by $F(u)=\int_a^b (u^2(x)-gu(x)-h)\,dx$ is continuous?
Thank you
You can decompose your functional $F=F_1-F_2-F_3$ where $F_1(u)=\| u\|_{L^2}^2$, $F_2(u)=(g,u)_{L^2}$ and $F_3(u)=h(b-a)$. Continuity of the first follows from that of the norm, the second is a bounded linear functional by Cauchy-Schwarz inequality and the third is constant.