Let $E := \mathscr{C}^1([0, 1], \Bbb R)$ with norm $∥ f ∥_{∞,1} := \sup{(| f(x)| + | f ′(x)| : x ∈ [0, 1])}$ and $F := \mathscr{C} ([0, 1], \Bbb R)$ with supremum norm $∥ f ∥_∞ := \sup(| f(x)| : x ∈ [0, 1])$. Show that figure $D : E → F : D( f ) := f'$ is continuous and determine the operator norm of $D$.
My idea would be to find a special function for the maximum and / or to estimate an optimal bound. However, im not sure about that.
Hint: By considering a sequence of sinusoids, you can get a sequence $f_k$ of functions for which $\sup |f_k'(x)|$ increases without bound but $\sup|f_k(x)|$ remains constant (as $k \to \infty$). What happens to $\|D(f_k)\|_\infty/\|f_k\|_{1,\infty}$ as $k \to \infty$?
Note also that $\sup(|f| + |f'|) \geq \sup |f'|$. What does this tell you about the operator norm?
Let us note a few technicalities regarding the linear maps and the operator norm. Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be normed vector spaces. In particular, $V$ is a metric space with the canonical metric $\|\cdot-\cdot\|_V$ and analogously for $W$. So, given a function $f : V \to W$, we may talk about if $f$ is continuous or not using the standard epsilon-delta definition of continuous maps between metric spaces (see the "Continuous functions between metric spaces" section). If we restrict our attention to linear functions $T : V \to W$, this notion of continuity becomes equivalent to showing that there exists some $c \in [0,\infty)$ such that $\|Tv\|_W \le c\|v\|_V$ for all $v \in V$ (see here). Given this definition, can you show that $D : E \to F$ is continuous? It may be helpful to note that $\|f\|_{\infty,1} = \|f\|_\infty+\|f'\|_\infty$ for $f \in E$. Here's two further hints (again please give this a try first).
To check this, let $f \in E$ be arbitrary. By definition, we want to find any $c \in [0,\infty)$ (independent of $f$) such that $\|Df\|_{\infty} \le c\|f\|_{\infty,1}$. Try expanding $\|Df\|_{\infty}$ and $\|f\|_{\infty,1}$ to find a possible $c$.
you should be able to prove continuity using $c=1$.
Now, the space of continuous linear maps from $V$ to $W$ is itself a vector space denoted $\mathcal{L}(V,W)$. For each $T \in \mathcal{L}(V,W)$ we know there exists some $c \in [0,\infty)$ such that $\|Tv\|_W \le c\|v\|_V$ for all $v \in V$. The infimum of all such $c$, lets call $c_T$, is a non-negative number we can associate to $T$. Indeed, the operator norm is $\|\cdot\|_{op}$ on $\mathcal{L}(V,W)$ is the defined to be $\|T\|_{op} = c_T$. It is a good exercise to prove that this is a norm. Now, there are may equivalent definitions of the operator norm, as this excellent Wikipedia page summarizes. The one that @BenGrossmann is referring to is
$$ \|T\|_{op} = \sup_{ v \in V\setminus \{0\} } \frac{\|Tv\|_W}{\|v\|_V}. $$
When proving $D : E \to F$ is continuous, the constant $c$ you found is an upper bound for $\|D\|_{op}$. Now, using the technique @BenGrossmann suggests, you should be able to find a lower bound as well. Together, the upper bound and lower bound should tell you what $\|D\|_{op}$ has to be. If after some trial and error you cannot find the functions $f_k$ that @BenGrossmann suggests, you may check these hints.
Sine waves are bounded in magnitude by 1. Yet, increasing the frequency of the waves can make the slopes of tangents to the sine waves as large as you like.
Let $k \ge 1$ be any integer. Take $f_k : [0,1] \to \mathbb{R}$ be the function $f_k(x) := \sin(kx/\pi)$. What is $\|f_k\|_{\infty}$? What is $f_k'$? What is $\|f_k'\|_{\infty}$? Note that $\|f_k'\|_{\infty}$ should increase as $k$ increases.
I will leave it to you though to complete the limiting argument. Feel free to edit your question with any progress you make and we can leave comments to help if you get stuck anywhere. Best of luck with your studies!
