Closed range is equivalent to a quotient estimate

Given: The spaces, maps, scalar field, and hypotheses in the statement above. All duals consist of linear functionals over the ambient field; evaluation has no conjugation.

From A quotient of a Banach space by a closed subspace is Banach, with its stated hypotheses: Assume the Axiom of Countable Choice (def-countable-choice). Let $X$ be a Banach space and let $M\le X$ be a closed linear subspace. Then $X\mathrm{/}M$ is Banach for the quotient norm.

From A bounded operator that vanishes on a subspace factors uniquely through the normed quotient, with its stated hypotheses: Let $X$ and $Y$ be normed spaces over the same scalar field, let $M\le X$ be a closed linear subspace, let $q:X\to X\mathrm{/}M$ be the quotient map, and let $T:X\to Y$ be a bounded linear operator with $M\subseteq \ker T$. Then there is a unique bounded linear operator $\bar{T}:X\mathrm{/}M\to Y$ such that $\bar{T}\circ q=T,$ and moreover $\mathrm{\parallel }\bar{T}\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$.

From A closed subspace of a Banach space is Banach, with its stated hypotheses: Let $V$ be a Banach space and let $W\subseteq V$ be a closed linear subspace, equipped with the restricted norm. Then $W$ is a Banach space.

From Bounded inverse theorem, with its stated hypotheses: Assume DC. A bounded bijective linear map $T:X\to Y$ between Banach spaces has a bounded linear inverse $T^{-1}:Y\to X$.

From Under Dependent Choice, a bounded operator between Banach spaces is bounded below exactly when it is injective with closed range, with its stated hypotheses: Assume the Axiom of Dependent Choice (def-dependent-choice). Let $X$ and $Y$ be Banach spaces over the same scalar field, and let $T:X\to Y$ be a bounded linear operator. Then $T$ is bounded below if and only if it is injective and has closed range.