Let $X$ be the space of all continuously differentiable functions on $[-1,1]$, with the norm $\|f \|_\infty$. On $X$ we also define the norm \|f \| = \|f \|_\infty + |f'(1)| for all $f$ in $X$.
- If $(f_n)_n$ converges to $0$ with respect to $\|f \|$, does it follow that it converges with respect to $\|f\|_\infty$?
- If $(f_n)_n$ converges to $0$ with respect to $\|f\|_\infty$, does it follow that it converges w.r.t. $\|f\|$?
For the first question I thought the answer is yes. And my proof is: if $f_n$ converges to $0$ w.r.t. $\|f\|$, then for every $\epsilon$, eventually \|f_n-0 \| = \|f_n \|_{\infty} + |f'_n(1)|<\epsilon and since \|f'_n(1)| \geq 0, we have |f_n-0|_{\infty} \leq |f_n-0| = \|f_n \|_{\infty}+ |f'_n(1)|<\epsilon. So, $(f_n)_n$ converges wrt $\|f\|_{\infty}$. But I am not sure if it is correct.
I would be glad if you could help me and give a hint for the second one (it seems obvious but probably I am missing something).
Thank you very much. And sorry, it is an easy question but I am new to analysis.