Cartan's theorems

Cartan's theorems are fundamental results in complex analysis and geometry. They describe how complex functions behave on certain spaces: Theorem 1 states that on a "nice" complex space, holomorphic functions can distinguish points and separate local structures, akin to functions having enough "resolution." Theorem 2 ensures that if the space has no holes or obstructions ("Stein spaces"), then the space of all such functions is as rich as possible, allowing complex analysis tools akin to those used on simple domains. Together, these theorems provide a deep understanding of the structure and function theory on complex spaces, guiding mathematicians in analyzing complex geometric shapes.