Remark

Two of the propositions proved earlier are consequences of this theorem: the continuity of the bilinear function $tr$, by virtue of the relation $xr = t \cdot x$; and Proposition 7, by applying Theorem 1 to the identity mapping of $E$, considered as a linear mapping of the space $E$, endowed with the norm $p$ into the space $E$ endowed with the norm $q$ (or vice versa).