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

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

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:

- 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:

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

And the strengthening:

- 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.

too many questionsput 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 bookGeneral Topology. $\endgroup$ – TaQ Mar 31 '15 at 9:47Too imprecise? $\endgroup$ – TaQ Mar 31 '15 at 9:53