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I know $\displaystyle\int_0^1 f(x)g(x) \, dx $ is an inner product for $ C[0,1]$ but I can't find a conclusive answer either way for $\displaystyle\int_0^{1/2} f(x)g(x) \, d x$

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    .... but o$n$ another hand, it is a semi-inner product.2012-12-06

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No, take $f(x)=\begin{cases}0 & \text{ if } 0\le x\le\frac{1}{2}\\x-\frac{1}{2}&\text{ if }\frac{1}{2}\lt x\le1\end{cases} $

Then $f\cdot f = 0$ but $f\neq 0$

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    ty, I've corrected it.2012-12-06