Identify $c/c_0$

Question

Let
$c=$ set of converging sequences
$c_0=$ set of sequences that converge to $0$.
Find a set isomorphic to the quotient space $c/c_0=\{ (a_n)+c_0: (a_n)\in c \}$

I have done a similar problem when $X$ is a hilbert space and $M$ is a closed subspace.
Where in that case the Quotient map $Q:X\rightarrow X/M$ that is $(Q(x)=x+M)$ restricted to the $M^{\perp}$ will become an isometric isomorphism.

But in the above mentioned question since they are not Hilbert spaces (Please correct me if I'm wrong)I cannot directly apply the quotient map to a perpendicular set. But I think there is an analogy.
Help would be appreciated

Answer 1

Consider the mapping $l\colon c\to\mathbb{R}$, $l((x_n)):=\lim_{n\to\infty} x_n$. This mapping is linear and surjective. The kernel equals $c_0$. Thus $c/c_0\sim l(c)=\mathbb{R}$.