Let $\bar C =$ be the double converges sequences ($(a_n)^\infty_{n=-\infty}$), and $\bar C_0$ be the double converges sequences where the limit is $0$ for both $-+\infty$. What is $\text{codim}(\bar C_0)$?
I know that in general, the codim would be $\dim(\bar C) - \dim(\bar C_0)$ but I cannot calculate the dimension in that case.
Any other approaches?
That is not the general definition of "codimension". The codimension of $X$ in $Y$ is the dimension of $Y/X$. So, we need to find the dimension of $\bar{C}/\bar{C}_0$.
Let $a = (a_n)_{n=-\infty}^\infty$ be the sequence such that $a_n = 0$ for all $n \leq 0$, and $a_n = 1$ for all $n > 0$.
Let $b = (b_n)_{n=-\infty}^\infty$ be the sequence such that $b_n = 0$ for all $n \geq 0$, and $b_n = 1$ for all $n < 0$.
Now, $a$ and $b$ are linearly independent, and so are their equivalence classes in $\bar{C}/\bar{C}_0$.
For any $c = (c_n)_{n=-\infty}^\infty \in \bar{C}$, let $L(c) = \lim\limits_{n \to \infty}c_n$ and $l(c) = \lim\limits_{n\to-\infty}c_n$. Then $c = L(c)a + l(c)b + (c - L(c)a - l(c)b)$, and \begin{align*}\lim\limits_{n\to\infty}(c_n - L(c)a_n - l(c)b_n) &= \lim\limits_{n\to\infty}c_n - L(c)\lim\limits_{n\to\infty}a_n - l(c)\lim\limits_{n\to\infty}b_n \\&= L(c) - L(c)1 - l(c)0 \\&= 0,\end{align*} and \begin{align*}\lim\limits_{n\to-\infty}(c_n - L(c)a_n - l(c)b_n) &= \lim\limits_{n\to-\infty}c_n - L(c)\lim\limits_{n\to-\infty}a_n - l(c)\lim\limits_{n\to-\infty}b_n \\&= l(c) - L(c)0 - l(c)1 \\&= 0,\end{align*} so $c$ differs from a linear combination of $a$ and $b$ by an element of $\bar{C}_0$, so in $\bar{C}/\bar{C}_0$, the equivalence class containing $c$ is a linear combination of the equivalence classes of $a$ and $b$. Thus, $\{[a],[b]\}$ is a basis for $\bar{C}/\bar{C}_0$, so $\mathop{\mathrm{codim}}(\bar{C}_0) = \dim(\bar{C}/\bar{C}_0) = 2$.