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Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm: $$ \Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r \vert \}^2, $$ where $T: \ell_2 \to \ell_2$ is a linear operator with $(Tx)_n = x_n/n$ and $r$ is any scalar. My question is: does this equivalent norm satisfy the Kadec-Klee property (i.e., the weak and norm convergence coincide on the unit sphere)? Kindly help me. Thank you.

Here, $\Vert (r, Tx) \Vert^2 = \vert r \vert^2 + \sum_{n=1}^\infty (\frac{x_n}{n})^2$ and we are working on the space $\mathbb{R} \times \ell_2$.

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  • $\begingroup$ If that's a homework question, it should be asked on math.stackexchange. Moreover, what is the first norm on the right hand side? Could you also state the question with using r just in terms of some x? $\endgroup$
    – Dirk
    Commented Aug 24 at 10:23
  • $\begingroup$ Thank you for your comment, sir. I have made the question clear in my recent edit. Also, it is not a homework question. This example can be found in the paper Mosco convergence and the kadec property-DOI- 10.1090/S0002-9939-1989-0969313-4. This equivalent norm does not have the dual kadec property, as shown in the paper. However, I am confused regarding whether this norm has kadec property on the space itself or not. $\endgroup$
    – PPB
    Commented Aug 24 at 11:08

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