I want to determine whether any of the two inner products
$\langle f, g \rangle=\int_{b}^{a}f'(x)g'(x)dx$ (1) and $\langle f, g \rangle=\int_{b}^{a}f'(x)g'(x)dx+f(a)g(a)$ (2)
can be inner products on $C^1[a,b]$, but am uncertain how to proceed.
$\langle f, g \rangle$ $\in$ $\left \{ \left. C^1([a,b])=f:[a,b] \to \mathbb{R} \vert f,f' {cont.} \right \} \right.$ and the elementary identities for the inner product must hold. Can I perhaps find a simple counter example?
The first one is not an inner product, the second one is. You should be able to verify linearity and symmetry by yourself. The only question is whether $\langle f , f \rangle=0$ implies $f=0$. In the first case we only get $\int f'(x)^{2}dx=0$ which means $f'(x)=0$ for all $x$ and $f$ is a constant. So we cannot conclude that $f=0$. In the second case we get $\int f'(x)^{2}dx=0$ and $f (a)=0$. Thus $f$ is again a constant and $f(a)=0$ so $f \equiv 0$.