Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
8  
Look at the definition of inner product again. –  Chris Eagle Oct 3 '12 at 10:59
1  
If you use wrong definitions, you can't complain that you get wrong conclusions :-) –  Siminore Oct 3 '12 at 11:05

1 Answer 1

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.