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)).



Reference request: Baire's theorem for operator ranges
(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.


