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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.

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    If you use wrong definitions, you can't complain that you get wrong conclusions :-)2012-10-03

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