Reference request: Baire's theorem for operator ranges

Question

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)).

Answer 1

(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.