We say that two norms $|| · ||_1$ and $|| · ||_2$ defined on the same vector space $F$ are equivalent if and only if there exist two constants $C > 0$ and $C '> 0$ such that for all $x ∈ F $, $C '||x||_2 ≤ ||x||_1 ≤ C||x||_2$.
Let $E = C ^1 ([0, 1])$. Let $f ∈ E$. We define $||f||_{L^∞} := \underset{x∈[0,1]}{\sup}|f(x)|$, $N(f) := |f(0)| + ||f ' ||_{L^∞}$ and $N '(f) := ||f||_{L^∞} + ||f '||_{L^∞}$.
Answer the following questions:
Show that $N(f)$ and $N '(f)$ are norms.
Show that $N(f)$ and $N '(f)$ are equivalent.
Show that $N(f)$ and $||f||_{^L∞}$ are not equivalent. [Hint: Assume that the two norms are equivalent. Then consider $f_n(x) := x ^n $ for $n ∈$ {$1, 2, ...$} ]
I have this problem as a homework assignment, I solved part 1, but I don't really know how to solve parts 2 and 3. We have never come across the term "equivalent norms" in the lecture, so I don't know how to do them. Please any help on how to work them out?
Since this is a homework problem I can only give hints.
For 2) use the fact that $|f(x)|=|f(0)+\int_0^{x} f'(t) dt| \leq |f(0)| +\|f'\|_{\infty}$. Taking sup over $x$ this gives $N(f) \leq N'(f) \leq 2N(f)$.
For 3) just note that $\|x^{n}\|_{\infty}$ is bounded and $\|nx^{n-1}\|_{\infty}$ is not. If there is a constant $C$ such that $N(f) \leq C\|f\|_{\infty}$ we get a contradiction by taking $f(x)=x^{n}$ in this inequality and letting $n \to \infty$.