# norms on vector space are equivalent iff they have same topology

Note that (1) all norms on finite dimensional vector space are equivalent.

And here (2) norms on vector space are equivalent iff they have same topology.

Question : How can we prove (2) ? Please recommend reference

Thank you in anticipation

Hint: think in the identity $$I:(E,\|\cdot\|_1)\longrightarrow(E,\|\cdot\|_2)$$ and in the zero-centered balls in both norms.
• I read statement (1) and (2) So in infinite dimensional vector space we has nonequivalent two norms. $\|\ \|_2$ cannot be a metric on $\mathbb{R}^\omega$ – HK Lee Nov 10 '15 at 11:24
• @HKLee, $\Bbb R^\omega$ is the set of eventually zero sequences or the set of all sequences? And what is your $\|\cdot\|_2$? – Martín-Blas Pérez Pinilla Nov 10 '15 at 11:28