Consider a vector space defined as all continuous functions $f:[-1,1] \to \mathbb{R}$ equipped with the following inner product $\langle f,g \rangle = \int_{-1}^{1} f(x)g(x) \ dx$
Now $\langle f,f \rangle = 0$ implies that $\langle f,f \rangle = \int_{-1}^{1} f(x)^2 \ dx = 0$
But $f$ doesn't have to be $0$ for the above to hold. Any symmetric function on $[-1,1]$ will have inner product $0$. How do we rectify this problem?