# Questions on the implications of Condition $(2)$ on $\{U_i,\alpha_i\}$ being an atlas for $X$ (image of $\alpha_i$ contained in Banach space)

Lang's "Fundamentals of Differential Geometry" Chapter 2.

Let $$X$$ be a set. Let $$\{U_i\}$$ be an Atlas of pairs $$(U_i,\alpha_i)$$ such that:

i) $$U_i \subset X$$ for all $$i$$ and $$\cup_i U_i = X$$

ii) Each $$\alpha_i$$ is a bijection of $$U_i$$ onto an open subset $$\alpha_i U_i$$ of some Banach space $$E_i$$, and for any indices$$i,j$$ we have that $$\alpha_j\alpha_i^{-1}: \alpha_i(U_i \cap U_j) \rightarrow \alpha_j(U_i \cap U_j)$$ is a $$C^p$$ isomorphism.

It goes on to say that the vector spaces $$E_i$$ do not need to be the same for all $$I$$, yet that by taking the derivative of $$\alpha_j\alpha_i^{-1}$$ we see that $$E_i$$ and $$E_j$$ are toplinearly isomorphic.

Thus I am confused because $$d(\alpha_j\alpha_i^{-1})$$ is only a linear homeomorphism from $$\alpha_i(U_i \cap U_j)$$ to $$\alpha_j(U_i \cap U_j)$$, which are both mearly contained in $$E_i$$ and $$E_j$$.... SO where is the linear homeomorphism from $$E_i$$ to $$E_j$$ which must exist if they are toplinearly isomorphic!!

(the morphisms in the toplinear category are linear contiuous maps between topological vector spaces)

The derivative $$d(\alpha_j\alpha_i^{-1})$$ is not a map from $$\alpha_i(U_i\cap U_j)$$ to $$\alpha_j(U_i\cap U_j)$$. Rather, it is a map from $$\alpha_i(U_i\cap U_j)$$ to the space of continuous linear maps $$E_i\to E_j$$: at each point of $$\alpha_i(U_i\cap U_j)$$, the derivative is some linear map $$E_i\to E_j$$. Moreover, since $$\alpha_j\alpha_i^{-1}$$ is assumed to be a $$C^p$$ isomorphism, its derivative must be invertible at each point, so the derivative at each point is a toplinear isomorphism $$E_i\to E_j$$.
Note, though, that you can only conclude that $$E_i$$ and $$E_j$$ are toplinearly isomorphic if $$U_i\cap U_j$$ is nonempty, so that there actually is a point of $$\alpha_i(U_i\cap U_j)$$ to take the derivative. As a result, the conclusion that the $$E_i$$ are all toplinearly isomorphic is only valid if $$X$$ is connected (so any two charts can be connected by a chain of overlapping charts).