I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.

  1. When is the norm a continuous function on $E$?

  2. When is the metric induced by the norm continuous on $E\times E$ (or equivalently, the weak topology coincides with the strong topology)?

Obviously, the second condition is stronger then the first. The former holds for example if $E$ is compact in the norm topology, or finite-dimensional. Of-course, the norm is continuous in these "tame" cases, but there is a very different example of this phenomenon -- the unit sphere.

It turnes out that for the Hilbert spaces these two conditions coincide: if a net $\left\{x_{i}\right\}_{i\in I}\subset X$ and $x\in X$ are such that $x_{i}\to^{w} x$ and $\|x_{i}\|\to \|x\|$, then $\|x_{i}-x\|\to 0$. This gives rise to yet another question:

  1. Which normed spaces have the property that conditions 1 and 2 coincide for every subset of them? Is it true for reflexive spaces?

Finally, I am interested in what kinds of topological spaces can we get by taking subsets of a normed spaces with the weak topology. There is a related criterion of compactness: a subset $E$ of a complete topological space $X$ is compact if and only if the uniformity of $X$ restricted to $E$ coincides with the uniformity of the weak topology on $X$ restricted to $E$. However, I am not sure if this fact is relevant.

Bounded subsets of a separable reflexive space are metrizable, and hence compactly generated (in the sense of general topology, not functional analysis).

From Krein-Smulian Theorem, any convex subset of a reflexive space $X$ is weakly closed if and only if its intersection with every weakly compact disk is closed. This condition is very similar to compact generatedness and so the question is:

  1. When is $E$ compactly generated? Is it true for $E=X$?

And the strengthening:

  1. When is $E$ locally compact?

I certainly do not expect comprehensive answers to my questions. However, references and examples would be very appreciated. Some information regarding question 3 is needed urgently.

Thank you.

PS: the question was completely reorganized in comparison with its original form, which was poor.

  • $\begingroup$ There are too many questions put together but for "I am interested if the Hilbert space itself is compactly generated ..." use: Every first countable (and hence in particular metrizable) or locally compact Hausdorff topological space is compactly generated. This is Theorem 13 on page 231 in Kelley's book General Topology. $\endgroup$ – TaQ Mar 31 '15 at 9:47
  • $\begingroup$ Or what topology your intend to have on "the Hilbert space itself"? Too imprecise? $\endgroup$ – TaQ Mar 31 '15 at 9:53
  • $\begingroup$ I am sorry, if I haven't state it clear enough, but everywhere I meant the weak topology. I guess it is not first countable on the Hilbert space, right? $\endgroup$ – erz Mar 31 '15 at 10:58

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