Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of $\mathbb H$? I know several examples where it is actually true, but the proof seems to depend on the particular structure of the norms on $\mathbb B, \mathbb H$, so I wonder if there is a general result of the sort.
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$\begingroup$ What does "embedded" mean here? Normally I'd think that there was a bounded linear map $\mathbb{B} \rightarrow \mathbb{H}$ which is also bounded below. But this definition would not make sense in the context of the rest of this question. $\endgroup$– Matthew DawsCommented Mar 12, 2021 at 13:31
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1$\begingroup$ It means only $\mathbb B\subset \mathbb H$ and for all $x\in \mathbb B$, $\Vert x\Vert_{\mathbb H}\le C\Vert x\Vert_{\mathbb B}$. $\endgroup$– BazinCommented Mar 12, 2021 at 14:02
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1$\begingroup$ An obvious situation is for reflexive Banach spaces because $T(\mathbb B)=\bigcup_{n\in\mathbb N} nT(B)$ for the unit ball $B$ of $\mathbb B$. Then $nT(B)$ are weakly compact and hence closed. $\endgroup$– Jochen WengenrothCommented Mar 12, 2021 at 14:26
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As mentioned in my comment, this is true for reflexive Banach spaces and the compactness game may generalize to other situations, e.g., if the Banach space is a dual space and the embedding in $\sigma^*$-$\sigma^*$-continuous. In general, the answer is no: A Banach space continuously included into a Hilbert space need not be $F_\sigma$ there:
Let $X=c_0$ be the usual Banach space of null sequences and $H$ a Hilbert space containing $c_0$, e.g., the space of all sequences $(x_n)_n$ such that $(x_n/n)_n \in\ell_2$. Assume that $X=\bigcup_n F_n$ with $H$-closed sets $F_n$. By Baire's theorem, some $F_n$ contains an $X$-interior point and hence some $X$-ball $B(x,\varepsilon)$. For the unit ball $B$ of $X$, this implies (by scaling and translation) that $D=\overline{B}^H$ is contained in $X$. The closed graph theorem for the inclusion of the linear span of $D$ with the Minkowski functional of $D$ as a complete norm (whose unit ball is $D$) implies that $D$ is bounded in $X$. Thus, $D$ is the unit ball of an eqivalent norm on $c_0$, and $D$ is $\sigma(H,H')$-compact since $H$ is Hilbert. But this is not true because an old theorem of Ng [On a Theorem of Dixmier. Math. Scand. 29 (1971), 279-280] says that a Banach space whose unit ball is compact in some coarser Hausdorff locally convex topology is a dual space.
The argument shows in fact that every Banach spcace which is embedded as an $F_\sigma$-set in some reflexive locally convex space is a dual space.
answered Mar 12, 2021 at 14:58
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3$\begingroup$ I guess that $\ell_\infty(S)$ with $S$ uncountable is the simplest example of a dual space that is not continuously embeddable into a reflexive Banach space. What is an example of a dual space that is continuously embeddable into a reflexive Banach space, but no continuous embedding into a reflexive space has $F_\sigma $ range? $\endgroup$ Commented Mar 12, 2021 at 16:17
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$\begingroup$ Thanks for that great and helpful answer. $\endgroup$– BazinCommented Mar 13, 2021 at 11:04
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$\begingroup$ @Jochen Wengenroth Would you have a reference for that theorem of Ng? Thanks in advance. $\endgroup$– BazinCommented Mar 15, 2021 at 15:18
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$\begingroup$ I have added the reference to the answer. In wikipedia,the theorem is called Dixmier-Ng theorem. $\endgroup$ Commented Mar 15, 2021 at 17:47