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Let X be an infinite-dimensional real Banach space with norm $\|\cdot\|$.

  1. Prove that there exists an unbounded linear functional $f: X\to \mathbb R$.
  2. Let $f$ be as above. Consider $T: X\to X$ given by $Tx=x-2f(x)x_0$ for each $x\in X$, where $x_0\in X$ satisfies $f(x_0)=1$. Prove that $\|Tx\|$ defines a complete norm on $X$ that is not equivalent to $\|\cdot\|$. (Hint: what is $T^2\,$?)

I've found that $T^2(x)=x$, for any $x \in X$. However, I don't know how to use this to prove $\|Tx\|$ is not equivalent to $\|\cdot\|$. Anyone who can help me? Thanks!

Frederik vom Ende
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Tony Weng
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  • Hint: use the fact that $f$ is unbounded. – GEdgar Apr 21 '23 at 15:46
  • If $x \mapsto |Tx|$ were equivalent to $x \mapsto |x|,$ then you'd find that $|Tx| \leq c |x|$ and this would entail something about the continuity of $T = I + c f.$ – William M. Apr 21 '23 at 16:05
  • For completeness sake: an answer to 1. can be found, e.g., here – Frederik vom Ende Apr 22 '23 at 07:52
  • At the risk of being mean, you have been asking questions with pretty much zero effort in trying to solve them. I understand that level of math is very difficult, but you still have to show a genuine attempt and spend a lot of time on it. Hard proofs aren't something any mathematician just provides for free. If you want to keep asking questions on this site, you have to prove to us you struggled through it, not just mention one step. – Accelerator Apr 23 '23 at 06:45

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