1
$\begingroup$

Let $N$ be a norm on a finite dimensional $\mathbb R$-vector space. I'd like to prove that $N$ is continuous with respect to the max-norm $\lVert\cdot\rVert_\infty$.

  1. I can prove continuity at zero by using a simple argument to show $N$ is bounded on the $\lVert\cdot\rVert_\infty$ unit sphere.
  2. After that, $\lim_{\epsilon\to 0} N(x+\epsilon)\leq N(x)$ follows by sublinearity.

But how can I get $\lim_{\epsilon\to 0} N(x+\epsilon)\geq N(x)$?

$\endgroup$
1
$\begingroup$

For any $\epsilon$ in the space, $$N(x)=N(x+\epsilon-\epsilon)\le N(x+\epsilon)+N(\epsilon)$$

$\endgroup$

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.