Why are we able to compare the open and closed string perspective in AdS/CFT? I am reading the book Gauge/Gravity Duality Foundations and Applications by M. Ammon and J. Erdmenger. In that book they state the relationship between parameters on both sides of the duality as
$$
g_{YM}^2=2\pi g_{s},  \;\;\;2g_{YM}^2N=\frac{L^4}{l_s^4}=2\lambda.
$$
They then derive the corrospondence by considering two cases.

*

*The Open string perspective
In this scenario they consider strings as small perturbations and therefore have to go to the limit where the string coupling is small $g_s\ll1$. Furthermore, they consider $N$ coincident D-branes, for which the effective coupling constant is given by $g_sN$ and therefore must have $\lambda\ll1$ for the expansion to be reliable.


*The closed string perspective
In the closed string perspective they consider D-branes as sources of the gravitational field, curving the surrounding spacetime. The characteristic length scale $L$, should be large to ensure weak curvature and thus have a reliable super gravity approximation. Since $L^4/l_s^4$ is proportional to $\lambda$, we must look at this in the limit of $\lambda \gg 1$.
We then find $\mathcal{N}=4$ SYM in the open string perspective and $AdS_5\times S^5$ in the closed string perspective. Finally the book says that the two perspectives should be equivalent descriptions of the same physics.
But how can that be? In one case we are looking at the limit of $\lambda \ll 1$ and in the other we consider $\lambda \gg 1$. How can these apparent opposite limits be equivalent descriptions of the same physics?
 A: String theories have two worldsheet co-ordinates and it can often be useful to interchange which ones we regard as space and time. I think the missing link you are looking for comes from exploiting that freedom. For a cylindrical worldsheet, something which looks like a closed string (blue) propagating with time also looks like a vacuum diagram of an open string (red).

In this sense, Ammon and Erdmenger are taking two theories already known to be dual and explaining how a Super Yang-Mills description of it becomes reliable as $\lambda \to 0$ while $AdS_5$ can be seen to emerge as $\lambda \to \infty$. I see that their figure 5.1 is similar in spirit to the above but IMO it is not emphasized enough.
This open-closed duality, which underlies the top-down development of AdS / CFT, has led to some highly non-trivial results. But I like to think about it by comparing two path integrals. One the one hand,
\begin{align}
\left < a \right | e^{-iHT} \left | b \right > = \int_{\phi(x, 0) = \phi_a(x)}^{\phi(x, T) = \phi_b(x)} e^{iS} \, [d\phi]
\end{align}
has no restriction in the $x$ direction so it describes a theory on $\mathbb{R} \times \{ \mathrm{time} \}$ which can be conformally mapped to a cylinder. We just specify configurations at $t = 0$ and $t = T$ to compute an amplitude. But there is a similar object
\begin{align}
Z = \int_{\phi(0, t) = \phi_a(t)}^{\phi(T, t) = \phi_b(t)} e^{iS} \, [d\phi]
\end{align}
which now does nothing to time so it's a partition function. But the states are no longer specified by functions on $\mathbb{R}$ but by functions on a strip of width $T$. This means we are dealing with a boundary QFT where $a$ and $b$ are spatial boundary conditions.
