6
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

Suppose that $p$ is a seminorm on $X$. Let $$ K=\{x\in X:\ p(x)=0\}. $$ Let $Y=X/K$, i.e. $Y$ is the vector space of the classes $x+K$. On $Y$, define $$\|x+K\|=p(x).$$ This is well-defined, because if $x_1+K=x_2+K$, this means that $p(x_1-x_2)=0$, so by the reverse triangle inequality $$ |p(x_1)-p(x_2)|\leq p(x_1-x_2)=0 $$ and so $p(x_1)=p(x_2)$. It is now easy to check that $\|\cdot\|$ is a norm on $Y$. If $\pi$ is the quotient map $\pi(x)=x+K$, then $\pi$ is linear and $$ p(x)=\|\pi(x)\|, \ \ \ x\in X. $$

$\endgroup$
  • 1
    $\begingroup$ God, Yes! I didn't know the map I was looking for was so close to home (facepalm). Thank you very much. $\endgroup$ – razorcherry May 21 '17 at 19:07
3
$\begingroup$

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 $\pi$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.