Quoting from wikipedia, we have
If $p$ is a seminorm on $V$, we write $V_p$ for the Banach space given by completing $V$ using the seminorm $p$. There is a natural map from $V$ to $V_p$ (not necessarily injective).
We will say that a seminorm $p$ is a Hilbert seminorm if $V_p$ is a Hilbert space, or equivalently if $p$ comes from a sesquilinear positive semidefinite form on $V$.
I am having trouble interpreting the above. First the completion of a vector space with respect to a seminorm is… what exactly?
Then, $V_p$ is supposedly a Hilbert space which would require positive definitness and not just semi-definiteness.
Do the authors mean that $V_p$ is the completion of $V / \mathrm{ker} \, p$ with respect to the norm induced by $p$?
Also, I have not managed to find any reference on this Hilbert space definition of nuclear spaces except for Blanchard, Brüning. But they use actual norms instead.
Best Answer
I finally found a reference in some lecture notes. Apparently, one does mean to define $V_p$ as the completion of $V/\mathrm{ker} \, p$ with respect to the norm induced by $p$.