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Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient argument, this is equivalent to the existence of a complete norm on $U$ such that the injection $U \hookrightarrow F$ is continuous.

Obviously, operator ranges are more general than closed subspaces, but they still have many nice properties of closed subspaces. For instance:

(1) Finite intersections and sums of operator ranges are operator ranges.

(2) Images and pre-images of operator ranges under bounded linear operators are, again, operator ranges.

(3) If $F$ is the algebraically direct sum of finitely many operator ranges $U_1, \dots, U_n$, then all these operator ranges are closed (and hence, the sum is also topologically direct).

(4) Every operator range in $F$ is either equal to $F$ or meager in $F$.

(5) In particular, by Baire's theorem, if $F$ is a union of countably many operator ranges $U_n$ ($n \in \mathbb{N}$), then one of the spaces $U_n$ is equal to $F$.

(Fun fact: Assertion (5) gives an abstract non-sense proof for the fact that there exists a function in $L^1(0,1)$ which is not in $L^p(0,1)$ for any $p \in (1,\infty)$.)

All these observations are rather straightforward to prove (maybe with the exception of (4), which is a consequence of the so-called little open mapping theorem). Many of these observations (along with several other nice results) can also be found in Section 2 of the 1980 paper "On the continuous image of a Banach space" by R. W. Cross. However, I could not find (4) and (5) in the literature.

Yet, it seems very likely that this is written down somewhere, and when using these observations I would rather like to give a reference instead of including a proof (although the proof is not difficult). Hence:

Question. I'm looking for a reference for (4) and (5) in an article or book.

(If I had the choice, I would prefer a reference which gives a somewhat comprehensive treatment of operator ranges in Banach spaces and thus also includes results such as (1)--(3); but currently, I don't know any reference at all from where I could quote (4) or (5)).

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  • $\begingroup$ Have you checked this paper? Cross, R. W.; Ostrovskij, M. I.; Shevchik, V. V., Operator ranges in Banach spaces. I. (English) Zbl 0834.47001, Math. Nachr. 173, 91-114 (1995). (Part II does not seem to exist.) $\endgroup$ May 20, 2021 at 16:36
  • $\begingroup$ @DirkWerner: Thank you for your comment! Yepp, I've check this paper before I posted the question (the paper was kindly mentioned by Mikhail Ostrovskii as an answer to an earlier question of mine; there he also confirms that there is no part II). I couldn't find (4) or (5) in this paper, though, and the focus of the article seems to be a bit different. $\endgroup$ May 20, 2021 at 16:41
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    $\begingroup$ Possibly relevant remark: if we restrict to separable Banach spaces, then operator ranges are analytic sets, and by a standard but nontrivial theorem of descriptive set theory, analytic sets have the property of Baire (BP). And using the Baire category theorem, it's pretty easy to see that a nonmeager linear subspace having the BP must have nonempty interior, and therefore equal all of $F$. This is probably overkill for you, though. $\endgroup$ May 20, 2021 at 19:12
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    $\begingroup$ Correction: the "pretty easy to see" fact is called Pettis' lemma and while the proof is short and elementary, it's maybe not exactly obvious except in retrospect. $\endgroup$ May 20, 2021 at 19:27
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    $\begingroup$ As a matter of historical reference, the results you require are written up in Buchwalter's thesis which is available online. The chapter of interest to you ("Espaces vectoriels bornologiques") is in Publ. Dép. Math. Soc. 6 (1965) 2-53--particularly relevant are the theorems 2,3.9, 2.7.3 and 2.7.7. This is couched in a different language (not referring to French rather than English!) and the central concept is that of a "disque complétant" ("Banach disk" in the answer of Jochen Wengenroth) which goes back to Waelbroeck (1961). $\endgroup$ May 21, 2021 at 4:22

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(4) follows immediately from a version of the open mapping theorem: If a continuous linear operator between between Banach (or Fréchet or even more general) spaces has non-meager range, then it is open and, in particular, surjective.

This is theorem 2.11 in Rudin's Functional Analysis. As you said, (5) is an easy consequence. A far more general result is Grothendieck's factorization theorem.

The linear continuous images of the unit balls of Banach spaces are called Banach discs. This is a standard notion in the locally convex theory. It is used, for example, to define ultrabornological locally convex spaces. Without checking, this shoud appear in the Introduction to Functional Analysis of Meise and Vogt and it is discussed certainly in Barrelled Locally Convex Spaces of Bonet and Perez-Carreras.

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  • $\begingroup$ Thanks a lot for the reference, and in particular for the additional information in your answer! $\endgroup$ May 21, 2021 at 13:08

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