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?