I am a bit confused about the notation A $\subset \subset B$ used in functional analysis. The definition I have says: $A \subset \subset B$ iff $A \subseteq B$ and $\bar{A}$ compact in $B$. Wikipedia on the other hand says that in addition to that $\bar{A}$ may not touch $\partial B$.
My question is now which definition is the correct one? And is the open unit ball $B_1(0)$ relative compact in the closed unit ball: $B_1(0) \subset \subset \bar{B}_1(0)$?
The notation $A \subset \subset B$ implicitly presupposes that $A$ and $B$ are subspaces of some Hausdorff space $X$, for example $X=\mathbb C^n$.
In that case it means that the closure $\overline A$ of $A$ in $X$ is a compact subset of $B$.
This is the usual interpretation, for example the one adopted by Hörmander's classic An introduction to complex analysis in several variables (cf. page xi).
So indeed you may write $B_1(0) \subset \subset \bar{B}_1(0)$ .
Note, however, that the notation $A \subset \subset B$ is usually used only when $A$ and $B$ are open subsets of $X$. In that case $\overline A$ is automatically disjoint from $\partial B$, just as Wikipedia claims.