Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came with the caveat that the statement does not hold for two arbitrary closed subspaces.

So, here's what I'm looking for:

Find a Banach space $X$ and closed subspaces $M,N$ such that $$ M+N=\{m+n\mid m\in M, n\in N\} $$ Is not closed in $X$.

Any references, hints, or answers are appreciated!

Let $X$ be a Hilbert space with orthonormal basis $\{e_n\}_{n\in\mathbb{N}}$. Let $x_n=e_{2n}$, and let $y_n=e_{2n}+\frac{e_{2n+1}}{n+1}$. Take $M$ to be the closed span of the $x_n$ and $N$ to be the closed span of the $y_n$. Note that $M+N$ contains $e_n$ for all $n$, so the closure of $M+N$ is all of $X$.

However, I claim that $M+N$ does not contain the vector $z=\sum \frac{e_{2n+1}}{n+1}$ and hence is not all of $X$. Indeed, if you could write $z=x+y$ for $x\in M$ and $y\in N$, it is clear that $y$ would have to be $\sum y_n$, since the only way to get a nonzero inner product with $e_{2n+1}$ when building an element of $M$ or $N$ is to use $y_n$. Since the sum $\sum y_n$ does not converge, there are no such $x$ and $y$.