True/False Real Analysis Qualifying Exam Questions The following is from a practice real analysis qualifying exam, and I had a couple questions about some of them.

$a$) I think this one is true, but I'm not sure how to prove it.
$b$) I know that $\partial^{\alpha}\hat{f} = \hat{[(-2\pi ix)^\alpha f]} (\xi)$. I don't necessarily see any reason that setting this equal to $0$ implies that $f=0$, so that would lead me to believe it might be false, but I'm unable to construct a counterexample.
$c$) We haven't done this yet, so skip.
$d$) True -- HW problem
$e$) Should be false if $d$ is true, but can't construct counterexample
$f$) I believe it's false and that $X=[0,1]$ with $f = \frac{1}{\sqrt x}\chi_{[0,1]}$ works.
$g$) I believe it's true using a comparison test argument
$h$) I believe it's false using $X = [0,1], M = \mathbb{B}_{X}, \mu_1 =$ Lebesgue measure, \mu_2 = $ counting measure  
So I primarily need help with a,b,e and just want to verify that my answers for d,f,g,h are correct. Thank you!

*

*$(a)$ is false, for a counterexample I'd suggest to look at an infinite dimensional space, say $\ell^2(\mathbb{N})$. (Recall that the unit ball is not compact)

*$(b)$ is true: the Fourier Transform maps Schwartz functions into Schwartz functions. The only constant Schwartz function is $0$. This also implies that $f = 0$ by Plancherel's equality, for example.

*$(e)$ Let's disprove the statement: assume the space is only complete wrt one of the two norms. If it is complete wrt $\|\cdot\|_2$ then it is complete wrt $\|\cdot\|_1$ using the inequality given, a contradiction. Then it must be complete wrt $\|\cdot\|_1$. Assume that it is possible to prove the other inequality, then it would be complete wrt $\|\cdot\|_2$ arguing as before.

*$(f),(g)$: you are correct.

*$(h)$: You can also take a look at this
