Invertibility and the inverse of the transpose

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 Surjectivity is equivalent to a lower bound for the transpose, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Assume DC. 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 Bounded below is equivalent to surjectivity of the transpose, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Assume DC. If $T:X\to Y$ is bounded linear between Banach spaces, then $(\mathrm{\exists }c>0\mathrm{\forall }x\in X:\mathrm{\parallel }Tx\mathrm{\parallel }\ge c\mathrm{\parallel }x\mathrm{\parallel })⟺T^{\ast }:Y^{\ast }\to X^{\ast }\text{ is onto}\mathrm{.}$

From Transposition reverses composition, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For bounded linear $T:X\to Y$, $S:Y\to Z$ between normed spaces, $(ST{)}^{\ast }=T^{\ast }{S}^{\ast },I_X^{\ast }=I_{X^{\ast }}\mathrm{.}$ For bounded $T,U:X\to Y$ and $a,b\in \mathbb{K}$, $(aT+bU{)}^{\ast }=a{T}^{\ast }+b{U}^{\ast }$.

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 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 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.