Jtdylktuy

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 $codim(bar C_0)$?

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 $L(c) = limlimits_{n to infty}c_n$ and $l(c) = limlimits_{nto-infty}c_n$. Then $c = L(c)a + l(c)b + (c - L(c)a - l(c)b)$, and begin{align*}limlimits_{ntoinfty}(c_n - L(c)a_n - l(c)b_n) &= limlimits_{ntoinfty}c_n - L(c)limlimits_{ntoinfty}a_n - l(c)limlimits_{ntoinfty}b_n \&= L(c) - L(c)1 - l(c)0 \&= 0,end{align*}
and
begin{align*}limlimits_{nto-infty}(c_n - L(c)a_n - l(c)b_n) &= limlimits_{nto-infty}c_n - L(c)limlimits_{nto-infty}a_n - l(c)limlimits_{nto-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 $mathop{mathrm{codim}}(bar{C}_0) = dim(bar{C}/bar{C}_0) = 2$.