Defining multiplication and differentiation in a dual space

Defining multiplication and differentiation in a dual space

Let $B$ be a space of complex-valued analytic functions from some compact
subset $C$ of $\mathbb{C}^n.$ $B$ is a Banach space with the uniform norm.
Now, the paper I am reading says that we would need the dual space $B'$ of
$B$. In this space, it says to "define differentiation and multiplication
by transposition."
Question: What does this mean?
I really have no idea. All I know is that $B'$ is the (Banach) space of
continuous linear functionals on $B$. I am also familiar with the
transpose of a linear map but I'm not sure how this allows us to define
multiplication and differentiation.
Are there alternative definitions/constructions I am not aware of?