If the conorm of a linear transformation is positive, is the transformation an isomorphism?

Question

We defined the conorm of a linear transformation as $\inf\{\frac{|Tv|}{|v|}:v\ne0\}$, where T is a linear map between norm spaces.

I have already proven that the conorm of an isomorphism is positive. However, I'm not sure about the converse.

I think that it should be true for finitely dimensional spaces, as it would imply that $\ker(T)=\{0\}$, and I know that for finitely dimensional spaces this happens iff T is an isomorphism.

Now, I don't really know much about spaces of infinite dimensions, so my question is whether there is a counterexample of that kind for which this would not be true?

Thanks for any help.

Related: https://math.stackexchange.com/q/4165201/169085 – Alp Uzman May 15 '22 at 13:03 — Alp Uzman, May 15 '22 at 13:03

score 2 · Answer 1 · answered Nov 20 '20 at 09:21

2

On $\ell^{2}$ define $T(a_n)=(0,a_1,a_2,...)$. Then the infmum is $1$ but $T$ is not surjective so it is not an isomorphism of $\ell^{2}$ onto itself. However the following is true:

If the infimum is positive and $T$ is bounded then $T$ is an isomorphism onto its range (which may be a proper sub space).

answered Nov 20 '20 at 09:21

Kavi Rama Murthy

311,013

Hello, I wanted to know what is $l^2$ in this example? – TdotA Nov 21 '20 at 21:25
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$\ell^{2}$ is the space of all sequences of real numbers $(a_n)$ under the norm $|(a_n)|=\sqrt {\sum_n |a_n|^{2}}$. @TdotA – Kavi Rama Murthy Nov 21 '20 at 23:14

If the conorm of a linear transformation is positive, is the transformation an isomorphism?

1 Answers1