TITLE: Is every seminorm induced by a linear operator into a normed space?
QUESTION [6 upvotes]: I'm reading an analysis textbook chapter on convex topological vector spaces, and there is this statement that (one of) the most common way(s) to define a topology on a vector space $X$ is by requiring the continuity of certain linear maps, (i.e via the semi-norms induced by given linear maps $T\colon X\to Y$ where $Y$ is a normed space). 
This makes sense, but then there's this that bugs me: "Can any semi-norm on $X$ be shown to be induced by some linear map T, to some normed linear space?" I can't find a counter-example to this, or show that it is true. Any insight would be appreciated.

REPLY [3 votes]: Let $p$ be a seminorm on a space $X$. Then $X/ \{x\in X\colon p(x)=0\}$ is a normed space under the norm $\|\pi(x)\|=p(x)$. Then the seminorm $p$ is implemented by the quotient map $\pi$.