the context of my question is finite element theory here, but my question is purely mathematical. Let $F$ be an affine mapping from a reference triangle $\hat{T}$ to a general triangle $T$, call it $F:\hat{T} \rightarrow{T}$ which has the form $$F(\hat{x})=B \hat{x} +b$$ for some matrix $B$ which maps a reference triangle $(0,0)$,$(1,0)$,$(0,1)$ into a general triangle $T$. This framework is classical in finite element books.
Let us consider a function $\hat{f} : \hat{T} \rightarrow R$, and a function $v$ defined as $$f = \hat{f} \circ F^{-1} $$
Equivalently, $\hat{f} = f \circ F$. We have the following 
(see here at page 63 for the lecture notes)
Question: I honestly can't understand how to get formula (4.7). I can obtain easily the formula before (4.7). In particular, I don't know how the term $\sum_i \sum_k b_{ij} b_{kl}$ can be bounded with the norm of $B$. I don't see how to make a norm appear from there. Any help is highly appreciated.