Inner Product Space (Continuously Differentiable Functions)

Question

Let $V=C^{2}[-\pi,\pi]$ be the space of real valued twice continuously differentiable functions defined on the interval $[-\pi, \pi]$. Set $$\langle f,g \rangle=f(-\pi)g(-\pi)+\int_{-\pi}^{\pi}f''(x)g''(x)dx$$ Is this an inner product on $V$?

Answer 1

Hint: the function

$$f(x)=x+\pi$$

belongs to $C^2([-\pi,\pi])$ and satisfies $f(-\pi)=0$, $f^{''}(x)=0$ for all $x\in\left[-\pi,\pi\right]$. This implies $\langle f,f\rangle=0$.