First of all, my English is very bad, sorry.
It is likely that we'll be asked something similar in an exam tomorrow, and I m totally not equipped to solve this exercise (i.e. I don't have the prerequisite) .
I'll sum this up, I don't know about mathjax or anyway to write mathematics symbols here yet, sorry.
${}$
Let $f$ be the most simple real function you will meet in an exam.
Is it in $L_2$?
Derivate it
Is it in $H^1(\mathbb{R})$ ?
Derivation again
Is it in $H^2(\mathbb{R})$ ?
There's no need to be very formal, I believe teacher just look at the answers and doesn't care much about explanations
Here's more details about how it's going
So there are two single variable real function $f_1 = 1$ when $x$ in $[0,1]$, and $0$ elsewhere $f_2 = \frac{1}{2}(x+|x|)$ for $x$ in $[-1,1]$ and $0$ elsewhere ( basically $x $ between $0$ and $1$ and $0$ elsewhere)
$1)$ Are these functions in $H^2(\mathbb{R})$ ?
OK I THINK
I think yes because there's no divergence for the sum (integration) of their square in their intervals
$2)$ Derivation of these functions. OK
_First one is a Dirac at $x=0$ minus a Dirac at $x=1$
_Second one is $1$ minus a Dirac at $x=1$
$3)$ are $f_1$ and $f_2$ in $H (\mathbb{R})$ ?
NOT OK
I don't even know what is $H$ !
I ended up thinking its some Hilbert space but maybe it's Sobolev's maybe it's Hardy's (doesn't look like it's this one tho)
So after some researches I tried to see if they have a norm well defined on theirs intervals but it doesn't look like it as Dirac can be dealt with that easily when integrated
There is a story about a test function and it is most likely this I should have learn about back when I had a few hours to think about it
$4)$ derivation of these functions again. NOT OK
I have no idea, I'm tempted to say that when you do the derivation of a Dirac it is a Dirac again but that is just instinct.
It looks like it works for $0$ because of the following
A Dirac at $0$ I think I can just say it is equal to $e^{i\frac{\pi}{2}}$ but the two minus Diracs at $x= 1$ got my number.
$5)$ are $f_1$ and $f_2$ in $H^2(\mathbb{R})$ ?
Even if I found the second derivatives I'd be clueless. Don't even know what $H$ is as there's no mention of it in the material the teacher put online.
Thanks for attention!
Edit : sorry for tagging it in group-theory