about Bounded Operator

Question

Let be $l_p$ ( $1 \le p \le \infty$ ) space of sequences in $\mathbb{C}$ or $\mathbb{R}$. How can I prove that operator $L(x_1,x_2,x_3,\ldots)=(a_{1}x_{1},a_{2}x_{2},a_{3}x_{3},\ldots)$ is bounded if and only if sequence $(a_{n})_{n}$ is bounded ?

Answer 1

Let $(a_n)_n$ be bounded and $x = (x_1, x_2, ...)$.

$$\vert \vert L(x) \vert \vert_p^p = \sum_{n=0}^\infty \vert a_ix_i \vert ^p \leq \vert \vert (a_n)_n \vert \vert_\infty^p \vert \vert x \vert \vert_p^p$$so $L$ is bounded.

Let $L$ be bounded and $e_i = (0, ...0, 1, 0, ...)$ the unit vector with a $1$ in its i-th component.

$$\vert \vert L \vert \vert \geq \vert \vert L(e_i) \vert \vert_p = \vert a_i \vert$$

so $(a_n)_n$ is bounded by $\vert \vert L \vert \vert$.