Banach completion of a seminorm quotient

Let $E$ be a vector space and $p$ a seminorm on $E$. Its kernel $\ker p=\{x:p(x)=0\}$ is a linear subspace, and $p$ induces a norm on the quotient vector space $E/\ker p$ by

The Banach completion of the seminorm quotient, denoted $E_p$, is the completion of this normed space. Thus $E_p$ is a Banach space, and the quotient map gives a canonical linear map $E\to E_p$ whose induced map from $E/\ker p$ is isometric and whose image is dense. Equivalently, $E_p$ is the complete normed space obtained by adjoining limits of all Cauchy sequences in $E/\ker p$.

When $q\geq p$ are seminorms, the identity on $E$ induces a contraction $E_q\to E_p$. These canonical linking maps are the ones used in the definition of a nuclear space.