Non-equivalent complete norms and would-be continuous linear functionals (Q1201345)

Collections of non-equivalent complete norms on real or complex vector spaces \(E\) are constructed using functionals in the algebraic dual \(E^*\). To be more precise, let \(\| \cdot\|\) be any complete norm on \(E\). For each \(0\neq a\in E\) and each \(\varphi\in E^*\) with \(\varphi (a)\neq 0\) a complete norm \(\|\cdot \|_ \varphi\) on \(E\) is defined by \(\| x\|_ \varphi= \inf_{t\in \mathbb{K}} \| x-ta\| +| \varphi (x)|\). This norm \(\|\cdot \|_ \varphi\) is equivalent to \(\| \cdot\|\) iff \(\varphi\in (E,\| \cdot\|)'\). Moreover, each \(\lambda\in E^*\) with \(\lambda (a)=0\) has equal functional norms \(\|\lambda \|_ \varphi= \|\lambda \|\), and hence is continuous in \(\|\cdot \|_ \varphi\) iff it is continuous in \(\|\cdot \|\). As a corollary it follows that \((E, \|\cdot \|_ \varphi)'= (E,\| \cdot\|)' \ominus \mathbb{K}\alpha \oplus \mathbb{K}\varphi\) for some \(\alpha\in (E,\| \cdot\|)'\) with \(\alpha (a) \neq 0\).