My question is motivated by the following little proposition:

**Proposition.** For a vector subspace $V$ of a Banach space $(X, \|\cdot\|_X)$ the following assertions are equivalent:

(i) There exists a Banach space $Z$ and a bounded linear operator $T: Z \to X$ with range $V$.

(ii) There exists a complete norm $\|\cdot\|_V$ on $V$ such that the canonical embedding of $(V, \|\cdot\|_V)$ into $(X,\|\cdot\|_X)$ is continuous.

(See below for a proof.)

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**Question.** Are (i) and (ii) also equivalent to the following assertion (iii)?

(iii) There exists a bounded linear operator $S: X \to X$ with range $V$.

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**Proof of the Proposition.** Obviously (ii) implies (i), so assume that (i) holds. Let $\tilde T: Z / \ker T \to X$ denote the injective operator induced by $T$; then $\tilde T$ also has range $V$. The inverse ${\tilde T}^{-1}$ is a closed linear operator $X \supseteq V \to Z / \ker T$, so $V$ becomes a Banach space with respect to the graph norm given by $\|x\|_V := \|x\|_X + \|{\tilde T}^{-1}x\|_{Z / \ker T}$ for all $x \in V$.

**Remark.** For Hilbert spaces results of this type can, for instance, be found in the paper "Fillmore and Williams: On Operator Ranges (1971)". In fact, the above proof is an adaptation of an argument that appears in the proof of Theorem 1.1 of this paper.