Let $\ell^{\infty}=\left\{(x_n)_{n\in\mathbb{N}}\subseteq \mathbb{C}\mid \sup_n\|x_n\|<\infty\right\}$ the space of bounded sequences.
Let $m\in\mathbb{N}, m>1$, be fixed. Why is the quotient space $\ell^{\infty}/\bigoplus\limits_{k=1}^m\mathbb{C}$ isomorphic to $\ell^{\infty}$?
I think, if there exists an isomorphism, then this isomorphism is induced by the identity $\ell^{\infty}\to \ell^{\infty}$, or that the isomorphism is something canonical... but I don't really get this.
Maybe you can try to construst a linear map $\ell^\infty\to\ell^\infty$ with kernel $\bigoplus\limits_{k=1}^m\Bbb{C},$ then the isomorphism got. An intuitive idea is to throw out the first $m$ components, which is a linear map. (Mind it's not the identity map.)