Definition of connection 1-forms - when we write $A(X^\#)=X$, is $A(X^\#)$ meant to be constant?

Question

Let $A$ be a connection $1$-form on a principal $G$-bundle and $X\in\mathfrak{g}$.

By the definition of connection $1$-forms, $A(X^\#)=X$. However, $A(X^\#)\in C^\infty(P,\mathfrak{g})$, so the equation doesn't make sense. Is $A(X^\#)$ assumed to be constant? That is, $$A(X^\#)_p=X$$ for all $p\in P$?