**Disclaimer:** When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is just due to my ignorance.)


**Question.** Let $E,F$ be Banach space and suppose that $E$ embeds densely and continuously into $F$ (so we consider $E$ as a subspace of $F$ from now on). Assume that there exists a constant $M \in (0,\infty)$ with the following property:


For each $f \in F$ we can find a sequence $(e\_n)$ in $E$ that converges to $f$ with respect to $\|\cdot\|\_F$ and that satisfies $\|e\_n\|\_E \le M \|f\|\_F$.


Does it follow that $E = F$?


**Remark.** I first thought the answer should be yes due to some application of the open mapping theorem: clearly, it suffices to show that $\|\cdot\|\_E$ and $\|\cdot\|\_F$ are equivalent on $E$, and by the open mapping theorem this is true iff $\|\cdot\|\_F$ is complete on $E$; but I wasn't able to prove that latter property.


Am I overlooking some simple argument, or a simple counterexample?


**Edit.** It is probably worthwhile to note the following fact:


As observed by Nate Eldredge in a (now deleted) comment, it is easy to see that the answer is "yes" if $E$ is reflexive: Given $f \in F$ and $(e\_n) \subseteq E$ as above, we can choose a subsequence of $(e\_n)$ that converges weakly (in $E$) to a vector $e \in E$. For each $f' \in F'$ this implies that $\langle f', f\rangle = \langle f', e\rangle$, so $f = e \in E$.



On dense embedding of Banach spaces
Great question! What you need is Sandy Grabiner's approximation lemma:


**Lemma**: *Let $E$ and $F$ be Banach spaces, let $T \in B(E,F)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [F]\_1$ there exists $x\_0 \in [E]\_M$ with $\|y - Tx\_0\| \leq r$. Then for each $y \in F$ there exists $x \in E$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.*


I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is: Theorem 3.35 of my book *Measure Theory and Functional Analysis*.


It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.


