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I asked this question on Math Stackexchange, but I didn't get an answer:

https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155

I came upon the following specific property for a (complete) normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:

If a sequence $x_n$ in $X$ satisfies $\displaystyle \lim_{n\to\infty}(||x_n+y||-||x_n||)=||y||$ for all $y\in X$, then $\displaystyle \lim_{n\to\infty}x_n=0$.

This is not true in $l_1$; take $x_n=e_n$, the unit vector basis. The same counterexample doesn't seem to work in $c_0$, $l_\infty$, and $l_p$ for $p\neq 1$. Is this actually true in these spaces, or is this property, in fact, never satisfied? I don't know if it makes a difference if we additionally restrict this property to $(x_n)_n$ bounded.

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  • $\begingroup$ I think one can check by an ultrapower argument that the version for bounded sequences is true for every uniformly convex Banach space. This argument might be a bit of an overkill, though. $\endgroup$
    – Jochen Glueck
    Commented Mar 16 at 20:54
  • $\begingroup$ Please see $\S$3 and $\S$4 in emis.de/journals/NYJM/j/2007/13-16.html . Also see the bottom of p. 391 in books.google.ca/… $\endgroup$
    – Onur Oktay
    Commented Mar 16 at 21:38
  • $\begingroup$ @OnurOktay While I see how this is connected with Kalton's property (L), I don't see a direct answer. Am I missing something? $\endgroup$
    – Markus
    Commented Mar 17 at 15:33
  • $\begingroup$ I just wanted to point out this connection that I thought might be useful. Sorry if I gave the wrong impression inadvertently. $\endgroup$
    – Onur Oktay
    Commented Mar 17 at 19:04
  • $\begingroup$ @OnurOktay It is indeed an intresting connection. Thanks for pointing it out, I wasn't aware of it. $\endgroup$
    – Markus
    Commented Mar 18 at 18:29

1 Answer 1

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The separable spaces that do not have your property for bounded $(x_n)$ (namely, there is a sequence in the unit sphere satisfying your limit condition) are characterised in the paper ``Thickness of the unit sphere, $\ell_1$-types, and the almost Daugavet property'' by V. Kadets, V. Shepelska and myself (Zbl 1235.46014; Houston J. Math. 37, No. 3, 867-878 (2011)).

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