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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 $V$?

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which properties should you verify to prove/disprove that?I mean, is the positive-definiteness which troubles you? – Avitus Nov 4 '13 at 18:31
up vote 1 down vote accepted

Hint: the function


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

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