I'm reading about the Fell topology and have a question on some preliminary material. My reference is these notes on automorphic representations. Let $G$ be a locally compact Hausdorff, second countable unimodular group with Haar measure $dg$. Let $(\pi,V)$ be a Hilbert space representation of $G$, continuous in the sense that for every $v \in V$, the map $g \mapsto \pi(g)v$ is continuous $G \rightarrow V$, where $V$ is given the norm topology.
For each $f \in L^1(G)$, we have a bounded linear operator $\pi(f)$ on $V$ given by $\pi(f)v = \int\limits_G f(g)\pi(g)v\space dg$. This makes sense as a vector valued integral. The operator norm $|| \pi(f)||$ is bounded by the $L^1$ norm $||f||_1$.
Let $\hat{G}$ be the unitary dual of $G$, the set of equivalence classes of irreducible unitary Hilbert space representations of $G$. For $f \in L^1(G)$, we define a seminorm by
$$||f||_{\ast} = \sup\limits_{\pi \in \hat{G}} ||\pi(f)||$$
The C-$\ast$ algebra of $G$ is defined to be the completion of $L^1(G)$ with respect to this seminorm.
What is meant by the completion of a vector space with respect to a seminorm? What are some good references on this topic?
Even without knowing that $\Vert \cdot \Vert$ is a norm, one can define the completion of $L^1(G)$ w.r.t. $\Vert \cdot \Vert$:
For any first-countable topological group, one can define its completion, using Cauchy-sequences.