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Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$ Is $\langle f,g \rangle$ an inner product space?

I'm just checking the four conditions from Kreyszig pg 129.

I have a few questions. I know the first few conditions are true but I'm unsure of the wording for my justification. Because these are continuously differentiable I do not need to incorporate any sort of measure or Lebesgue integral, correct? So the scalar removes due to properties of the Riemann integral constructed as partial sums? Is this a real vector space? The functions are real valued, so is the inner product Hermitian symmetric from this fact?

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If you set the boundary value to be 0, then it is. – Shuhao Cao Jun 10 '13 at 2:36
up vote 8 down vote accepted

If $\;f(x)=c=$ a constant, then $\;\langle f,g\rangle=0\;\;\forall\,g\;$ , but $\,f\neq 0\;$ ...

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