Any Examples Of Unbounded Linear Maps Between Normed Spaces Apart From The Differentiation Operator? Let $X$ and $Y$ be normed spaces, either both real or both complex; let $T \colon X \to Y$ be a linear mapping. If there is a real number $r > 0$ such that 
$$ \lVert T(x) \rVert_Y \leq r \lVert x \rVert_X \qquad \mbox{ for all } x \in X, $$
then $T$ is said to be a bounded linear operator. 
One example of this operator is the following: 
Let $X$ denote the normed space of all the real polynomials defined everywhere on the closed interval $[0, 1]$, with the maximum norm, and let $T \colon X \to X$ be the map $x \mapsto x^\prime$. Then this linear operator is unbounded. 
Can we find any other examples of unbounded linear operators? 
I know that every linear operator whose domain is a finite-dimensional normed space is bounded. 

If $X$ is an infinite-dimensional normed space, then you can choose a Hamel basis $\{ x_{\alpha} \}$ and define a linear functional $F$ by setting $F(x_{\alpha})=s_{\alpha}$, where $\{ s_{\alpha} \}$ is a set of scalars. $F$ will be unbounded if $\sup_{\alpha} |s_{\alpha}|/\|x_{\alpha}\|=\infty$. Then you can extend the argument to obtain $T: X \rightarrow Y$ by choosing a finite set of discontinuous functionals $F_n$ on $X$, choosing an independent set of vectors $\{ y_n \} \subset Y$, and then setting $F(x)=\sum_n F_n(x)y_n$.
