Banach closed-range theorem

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.

The stated Axiom of Choice is the direct premise for the AC-qualified range, annihilator, and dual-quotient suppliers below (The Axiom of Choice).

The Axiom of Choice implies Dependent Choice, so the DC-qualified suppliers below apply (AC implies DC implies countable choice).

From Closed range is equivalent to a quotient estimate, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Assume DC. For a bounded linear map $T:X\to Y$ between Banach spaces, $\mathrm{ran}T\text{ is norm closed}⟺\mathrm{\exists }C>0\mathrm{\forall }x\in X:\mathrm{dist}(x,\ker T)\le C\mathrm{\parallel }Tx\mathrm{\parallel }\mathrm{.}$

From Membership in the transpose range by an operator estimate, under the stated Axiom of Choice: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Let $T:X\to Y$ be bounded linear between normed spaces and $f\in X^{\ast }$. Then $f\in \mathrm{ran}{T}^{\ast }⟺\mathrm{\exists }C\ge 0\mathrm{\forall }x\in X:\mathrm{\mid }f(x)\mathrm{\mid }\le C\mathrm{\parallel }Tx\mathrm{\parallel }\mathrm{.}$ For any such $C$, a representing $g\in Y^{\ast }$ can be chosen with $\mathrm{\parallel }g\mathrm{\parallel }\le C$.

From Surjectivity is equivalent to a lower bound for the transpose, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Assume the Axiom of Choice. For a bounded linear $T:X\to Y$ between Banach spaces, $T\text{ is onto}⟺\mathrm{\exists }C>0\mathrm{\forall }g\in Y^{\ast }:\mathrm{\parallel }g\mathrm{\parallel }\le C\mathrm{\parallel }{T}^{\ast }g\mathrm{\parallel }\mathrm{.}$

From Elementary kernel and range annihilator identities, under the stated Axiom of Choice: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a bounded linear $T:X\to Y$ between normed spaces, $(\mathrm{ran}T{)}^{\perp }=\ker T^{\ast },{}^{\perp }(\mathrm{ran}{T}^{\ast })=\ker T,{\overline{\mathrm{ran}T}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}={}^{\perp }(\ker T^{\ast })\mathrm{.}$ The closure in the last identity is in $Y$.

From The dual of a closed subspace is a dual quotient, under the stated Axiom of Choice: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Let $X$ be normed and $M\le X$ closed. Restriction $R:X^{\ast }\to M^{\ast }$ induces a linear isometric bijection $\tilde{R}:X^{\ast }\mathrm{/}{M}^{\perp }⟶M^{\ast },f+M^{\perp }⟼f{\mathrm{\mid }}_M\mathrm{.}$ Also $\mathrm{\parallel }R\mathrm{\parallel }\le 1$; its norm is $1$ when $M\mathrm{\ne }\{0\}$ and $0$ when $M=\{0\}$.

From Distance to an annihilator is the restriction norm, under the stated Axiom of Choice: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. If $M$ is a closed linear subspace of a normed $X$ and $f\in X^{\ast }$, then $\mathrm{dist}(f,M^{\perp })=\mathrm{\parallel }f{\mathrm{\mid }}_M\mathrm{\parallel }\mathrm{.}$

From Annihilators and preannihilators are norm closed, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For any normed $X$ and arbitrary $M\subseteq X$, $N\subseteq X^{\ast }$, both $M^{\perp }\subseteq X^{\ast }$ and ${}^{\perp }N\subseteq X$ are norm-closed linear subspaces. Moreover ${\bar{N}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}\subseteq ({}^{\perp }N{)}^{\perp }$.

From The transpose is bounded with the same norm, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a bounded linear $T:X\to Y$ between normed spaces, $T^{\ast }:Y^{\ast }\to X^{\ast }$ is bounded linear and $\mathrm{\parallel }{T}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$.

From If (Y) is Banach then (\mathcal B(X,Y)) is Banach, with its stated hypotheses: Let $X$ and $Y$ be normed spaces over the same scalar field. If $Y$ is Banach, then $\mathcal{B}(X,Y)$ is Banach for the operator norm.

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.