Boundedness of the differentiation operator

Prove that the operator $A: C^{1}(\left[a, b\right]) \rightarrow C(\left[a, b\right])$, which is defined by the formula $(A x)(t) = \frac{d x(t)}{d t}, t \in [a, b]$, is linear and continuous. Find its norm.

Let $X, Y$ be linear normed spaces. An operator $A: D(A) \subset X \rightarrow Y$ is said to be densely defined if $\overline{D(A)} = X$, and closed, if for any sequence $\left\{ x_{n}: n \geq \right\}$ 1) $\subset X$ such that $x_{n} \rightarrow x$ in $X$ and $A x_{n} \rightarrow y$ in $Y$ it follows that $x \in D(A)$ and $y = A x$.

Prove that the operator $A: C^{1}(\left[a, b\right]) \subset C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$, where $(A x)(t) = \frac{d x(t)}{d t}, t \in [a, b], x \in C^{1}(\left[a, b\right])$, is densely defined and closed, but not continuous.

from $C^{n}[0,1]$ to $C^{n-1}[0,1], n = 1,2, \ldots ;$

from $W_{2}^{1}[0,1]$ to $L_{2}[0,1]$.

Define $D: C_{b}^{1}(\mathbb {R}) \rightarrow C_{b}(\mathbb {R})$ by $D f = f^{\prime }$, and show that $D$ is a bounded operator.

Let $D: C_{b}^{1}(\mathbb {R}) \rightarrow C_{b}(\mathbb {R})$ be the same operator as in part (a), but replace the norm on $C_{b}^{1}(\mathbb {R})$ by the $L^{\infty }$-norm. Show that $D$ is now an unbounded operator.

$A(x) = x(t)$;

$A(x) = x^{\prime }(t)$.

Show that $C^{1}([0,1])$, with the supremum norm, is a subspace of $C([0,1])$ but that it is not closed.

Prove that the differential operator

is linear but not bounded.