Completing a vector space WRT $2$ different norms

Question

Let $V$ be a vector space, endowed with two norms $\|\,\|_1$ and $\|\,\|_2$, and denote by $V_1$, and $V_2$, the respective completions of $V$. If $$ \|v\|_1 \leq \|v\|_2, \textrm{ for all } v \in V, $$ then what can we say about the relationship between $V_1$ and $V_2$? If $\{v_n\}_n$ is a Cauchy sequence WRT $\|\,\|_2$, then it must be a Cauchy sequence WRT $\|\,\|_2$, because of the inequality. Moreover, if two Cauchy sequences are WRT $\|\,\|_2$, then they are equivalence WRT $\|\,\|_1$. This gives us a not necessarily injective map $$ V_2 \to V_1. $$ Is it possible that this map is sometimes not surjective?

Answer 1

$\newcommand{nrm}[1]{\left\lVert {#1}\right\rVert}\newcommand{\norm}{\nrm\bullet}$Yes, and it happens all the time. For instance, consider $V=P[0,1]$ the real vector space space of restrictions of polynomial functions to the interval $[0,1]$, with norms $\norm_2$ and $\norm_\infty$. It is clear that $\nrm{p}_2\le \nrm{p}_\infty$ for all $p\in P[0,1]$. With your procedure this inequality gives the tautological embedding of completions $(C[0,1],\norm_\infty)\hookrightarrow (L^2[0,1],\norm_2)$. The image of this embedding is, of course, a dense subspace of the codomain.