Definition 11

Let $u$ be a partial operator on E. A partial operator $v$ on E is said to be bounded relatively to $u$ if dom($u$)$\subset$ dom($v$) and if $v$ defines, by passage to the subspace, a continuous linear mapping of $E_u$ into E.

Let $u$ be a partial operator on E. Let $v$ be a partial operator bounded relatively to $u$. There exists a real number $m$ such that

for all $x\in$ dom($u$). The greatest lower bound of the real numbers $a\geqslant 0$ such that there exists a real number $b$ such that

is called the relative norm of $v$ with respect to $u$, and is denoted by $\|v\|_u$,

for every $x\in$ dom($u$). The relative norm of $v$ is therefore less than or equal to the norm of the restriction of $v$ to the space $E_u$.