Let $\mathcal{H}$ be the vector space of all complex-valued, absolutely continuous functions on $[0,1]$ such that $f(0)=0$ and $f^{'}\in L^2[0,1]$. Define an inner product on $\mathcal{H}$ by $\langle f,g\rangle=\int_0^1f^{'}(x)\overline{g^{'}(x)}dx $ for $f,g\in\mathcal{H}$.
If $0
I was able to show $L$ is linear. That was easy. I am having trouble showing it is bounded and I cannot determine what $\|L\|$ is.