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.

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  • $\begingroup$ This is for finite dimensional case ? $\endgroup$ – HK Lee Nov 10 '15 at 11:18
  • $\begingroup$ @HKLee, is for the question in the title. $\endgroup$ – Martín-Blas Pérez Pinilla Nov 10 '15 at 11:19
  • $\begingroup$ I read statement (1) and (2) So in infinite dimensional vector space we has nonequivalent two norms. $\|\ \|_2$ cannot be a metric on $\mathbb{R}^\omega$ $\endgroup$ – HK Lee Nov 10 '15 at 11:24
  • $\begingroup$ @HKLee, I don't understand your second comment. $\endgroup$ – Martín-Blas Pérez Pinilla Nov 10 '15 at 11:26
  • $\begingroup$ @HKLee, $\Bbb R^\omega$ is the set of eventually zero sequences or the set of all sequences? And what is your $\|\cdot\|_2$? $\endgroup$ – Martín-Blas Pérez Pinilla Nov 10 '15 at 11:28

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