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Aug 24 at 11:08 comment added PPB 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.
Aug 24 at 10:55 history edited PPB CC BY-SA 4.0
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Aug 24 at 10:23 comment added Dirk 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?
Aug 24 at 10:02 history asked PPB CC BY-SA 4.0