Fréchet derivative; open set

Question

I have some problems in understanding the definition of the Fréchet derivative of an operator $F: X \rightarrow X$. In fact, most authors report that $F$ must be defined in some open neighborhood of the point $x_0$ at which the derivative is computed. However, I would like to know: open with respect to what norm? Assume that $F$ is defined on some domain $D$ which is endowed with some norm that is different from the norm in $X$. Then, the neighborhood of $x_0$ should be open with respect to which of the two norms?

Answer 1

The Fréchet derivative is defined for maps $f$ from a non-empty open subset $A$ of a normed space $X$ into another normed vector space $Y$. Here, the norm we use on $A$ is the one inherited from $X$.