About the positivity of the inner product on $L^2[0,1]$

Question

My textbook on Hilbert space theory claims that the map $$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$ is an inner product on $C[0,1]$. But I am not sure whether $\langle\cdot,\cdot\rangle$ is positive. For the functions $f(x)=1$ and $g(x)=-1$ are both continuous, but $$\langle f,g\rangle=\int_0^1-1~\mathrm{d}x=-1<0.$$

I know that I must be overlooking something, but I cannot find what. Can someone help me? Thanks in advance.

Answer 1

Positivity means $$ \left(\forall f \right) \left(f \in L^2([0,1]) \Rightarrow \langle f,f \rangle \geq 0 \right) $$