Recall from Theorem VII.2.19 of Helemskii's monograph "The homology of Banach and Topological Algebras" that amenability of a Banach algebra $A$ is equivalent to any of the conditions below:

(i) derivations into dual modules are inner,

(ii) $\mathcal{H}_1(A,X)=0$ and $\mathcal{H}_0(A,X)$ is Hausdorff for all $A$-bimodules $X$.

The condition $\mathcal{H}_0(A,X)$ means that the image of the map

$d_0\colon A\widehat{\otimes}X\to X,\,\,\,\,d_0(a\otimes x):=a\cdot x-x\cdot a$

is closed. Now, when proving $(i)\Rightarrow(ii)$ the closedness of $\operatorname{im}d_0$ follows by a general fact (see Lemma 0.5.1 in this same Helemskii's monograph). My question is: is it possible to show closedness of $\operatorname{im}d_0$ directly?

The reason for a direct proof lies in the fact that I am working beyond Banach algebra category where -- in particular -- Open Mapping Theorem is not available.