Is there example for isomorphic closed subspaces of a Banach space with non isomorphic quotient?

Question

$Y_1$ and $Y_2$ are closed subspaces of a Banach space X and $Y_1 \simeq Y_2$. I can't find a way to show $X/Y_1 \simeq X/Y_2$ and it made be think that it's not true. Is there a counter example?

Answer 1

Let $X=\ell^2(\mathbb{N})$, let $Y_1=X$, and let $Y_2=\{(0,x_2,x_3,\ldots)\in\ell^2\}$ (the subspace of sequences whose first entry is $0$). Then $Y_1\cong Y_2$ - specifically, the right-shift map $S_r:Y_1\to Y_2$ is an isomorphism - but $X/Y_1\cong 0\not\cong \mathbb{R}\cong X/Y_2$.

A similar trick will show for many sorts of mathematical objects that isomorphic sub<object>s do not necessarily produce isomorphic quotient <object>s. For example, in the abelian group $M=\prod_{i=1}^\infty\mathbb{Z}$, the subgroups $N_1=M$ and $N_2=\{(0,x_2,x_3,\ldots)\in M\}$ are isomorphic while $M/N_1\cong 0\not\cong \mathbb{Z}\cong M/N_2$.