$C_{0}(\mathbb{R})$ is not Hilbert space.

Question

The space $C_{0}(\mathbb{R})$ of all complex valued continuous function that vanish outside some finite interval is not an Hilbert space under the inner product

$$(f,g)=\int_{-\infty}^{\infty} f(x)\overline{g(x)}\, dx$$

I tried to find out functions such that they violate parallelogram identity, but can't find out. Need help.

Answer 1

And you won't find them, since $(\cdot,\cdot)$ is an inner product. The problem lies elsewhere: your space is not a complete metric space.