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Given

$\left< f,g \right> =\int _{ 0 }^{ 1 }{ f'(t)g(t)dt } \quad in\quad C\left[ 0,1 \right] $

How can I prove if this is (or not) an inner product on the given vector space?

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    Check the [axioms](https://en.wikipedia.org/wiki/Inner_product_space#Definition). – 2017-02-27
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    I think the real tipoff for this one is that this "inner product" isn't even defined for all $f,g \in C[0,1]$. – 2017-02-27

1 Answers 1

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Hint Consider a nonzero constant function $f$ and compute $\langle f,f\rangle$.

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    So, for example, if f=1, $\int _{ 0 }^{ 1 }{ 0\cdot 1dt=0 } $ Right? Thanks a lot! Now I see it was a silly question, but i'm not used to this – 2017-02-27
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    I could not see the problem as the calculation of the inner product is 1 with me, and also why $C_{2}[0,1]$ is an inner product space with the inner product defined by $$ = \int_{0}^{1}f(x)\bar g(x) dx$$, what is the difference? – 2017-10-23