Surjection of norms Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that
\begin{equation*}
\| \cdot\|_1 \leq \| \cdot \|_2.
\end{equation*}
Is it true that there always exists a surjection between the two different completions of $V$? That is, does there always exist a continuous $\mathbb{C}$-linear surjection ($\mathbb{R}$-linear surjection)
\begin{equation*}
\overline{V}^{\|\cdot \|_2} \twoheadrightarrow \overline{V}^{\|\cdot \|_1} \, ?
\end{equation*}
The particular example I have in mind is when $G$ is an infinite discrete countable group and $V = \mathbb{C}[G]$, the complex valued functions on $G$ with finite support. Then there is a surjection from the maxmial group $C^*$-algebra and the reduced $C^*$-algebra.

Yes. Because you consider the identity map $(V,\|\cdot\|_2)\longmapsto(V,\|\cdot\|_1)$. The inequality between the norms shows that it is bounded. 
It is not immediately obvious to me that it is a surjection in the generality of your question. But it is in the concrete case you mention, because a C$^*$-homomorphism has closed image (when the domain is a C$^*$-algebra).
