This is an exercise from a book called "theory of complex functions" I want to solve:
Let $c\in G$ and $B\subset G$ a disc around c. When is the $\mathbb{C}$-algebra homomorphism $\mathcal{O}(G) \rightarrow \mathcal{O} (B)$ defined by $f\mapsto f_{|B}$ one-to-one? When is it onto?
I am not sure what this exercise asks. The $\mathbb{C}$-algebra maps a function to a function with its ensemble of definition restricted to the map of its disc? Or a domain is mapped to a disc in the domain?
If the second is the case, then I would say that it is never one-to-one, because the ensemble of definition has a higher cardinality than the ensemble of the map, but always onto.
Merci for any explaination.