Consider the normed vector space $c_{00}(\mathbb N ) = \{ x=(x_n) \in \mathbb{R} ^{\mathbb N} : \{ x_n \neq 0 \} \quad \text {is finite} \}$. Let $Id$ the identity operator. Let $ X =( c_{00} , \| \cdot\|_1)$ and $ Y= (c_{00} , \| \cdot\|_2)$, where $\|\cdot\|_p$ is the $\ell^p$-norm $p=1,2$; i.e. $\|(x_n)\|_p=\left(\sum |x_n|^p\right)^{1/p}$ $p=1,2$.
(a) Does $Id \in \mathcal{B}(X,Y)$ ?
(b) Does $ Id \in \mathcal{B}(Y,X)$ ?