Codimension and sequences Let $c$ be the set of all convergent sequences and $c_0$ the set of sequences tending to zero. How would one find the codimension of $c_0$? In particular, how would one find $\text{dim} 
\ c/c_0$?
So one needs to choose some arbitrary sequence in $c_0$ and represent it as a linear combination of elements in $c/c_0$?
 A: Your final paragraph makes no sense. The elements of $c/c_0$ are equivalence classes of sequences in $c$: they are elements of the form $[a] = \{b\in c\mid a-b\in c_0\}$ where $a$ is some element of $c$. You cannot express an element of $c_0$ as a linear combination of elements of $c/c_0$, because they are not even the same kind of things: the elements of $c_0$ are sequences, the elements of $c/c_0$ are sets of sequences.
The simplest way to find the codimension is to find an onto linear transformation from $c$ to some vector space $\mathbf{V}$ with kernel exactly $c_0$; the codimension will be dimension of $\mathbf{V}$. 
This is easy in this case: you can either work abstractly, directly with $c/c_0$, or try to find the map. Herer the former works: when will $a,b\in c$ be congruent modulo $c_0$? If and only if $a-b\in c_0$, if and only if $\lim(a-b)=0$, if and only if $\lim a = \lim b$. The equivalence classes correspond to the limits of the sequences. This suggests exactly which map to pick: let $T\colon c\to\mathbf{F}$ be the map to the underlying field given by $T(a) = \lim a$. It is straightforward to see that $T$ is linear, and that $\ker(T) = c_0$. Thus, $\mathbf{F} \cong c/c_0$. Since the dimension of $\mathbf{F}$ is $1$, the dimension of $c/c_0$ is $1$, so the codimension of $c_0$ in $c$ is $1$.
