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Let $A: L^2(\mathbb R) \to L^2(\mathbb R)$ be a linear operator with domain $domain(A) = H^2(\mathbb R)$ and $A = \frac{d^2}{dx^2}$.

The book says that $A$ is injective but I cannot show this.

I tried as follows:

$A$ is injective iff $Af = 0$ then $f =0$ holds.

But $Af = 0$ means $\frac{d^2}{dx^2} f(x) = 0$.

Then $f(x)$ can be constant or something like $f(x) = x$ so I could not show that $f=0$.

Would you please give me a comment for this?

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    What constant or linear functions are in $H^2(\Bbb R)$? – 2017-01-14
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    @Arthur That sounds like an answer to me. – 2017-01-15
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    @Arthur thanks for the comment, but would you give me some more detail? – 2017-01-15
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    @M.Doe what's the requirement for a function to be in $H^2(\mathbb{R})$? do constant or linear functions satisfy it? – 2017-01-15
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    You say you worry about functions like $f(x) = 3$ or $f(x) = x$, but you've forgotten to take into account that this isn't $C^2(\Bbb R)$, this is $L^2(\Bbb R)$. – 2017-01-15

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We have $f' = \text{cst}$. But since $f'\in L^2$, this constant must be zero. So $f=\text{cst}$. Again $f\in L^2$, so the constant must be $0$.