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A normed space $E$ has a

  • Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.

  • Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also norm-convergent (this seems to be the same as Radon-Riesz property, if I am not mistaken).

  • Dual Kadec (-Klee) property if the norm and weak* topologies (sequence convergences) coincide on the unit sphere of $E^{*}$.

I am not a Banach-space theorist, and so these properties seem somewhat intangible for me, so for writing purposes, I would like to know

more readily verifiable condition on the space $E$, that would imply that $E^{*}$ has a Kadec (or at least Kadec-Klee) property.

I was able to find that any locally uniformly rotund space has a Kadec-Klee property. $E$ is said to be locally uniformly rotund (LUR) if $\lim\|x_n+x_0\|=2\lim x_n=2\|x_0\|$ implies that $x_0=\lim x_n$. If $E^*$ is LUR, then the norm of $E$ is Frechet-differentiable, but the converse seems to fail. A uniform smoothness of $E$ implies that $E^*$ is uniformly rotund, and so LUR, and hence KK, but this seems to be too restrictive.

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  • $\begingroup$ I think that you need to clarify: What do you mean by a "more readily verifiable condition"? $\endgroup$
    – August Cleaner
    Commented Apr 9, 2017 at 17:32
  • $\begingroup$ @AugustCleaner: I don't really mean anything specific. Maybe something quantitative like some strong smoothness of the norm, or maybe some reflexivity-like condition. Anything would do, I just get a feeling that the uniform smoothness is too much. $\endgroup$
    – erz
    Commented Apr 9, 2017 at 23:01

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