Closed range theorem

(Bonsall [1986]) Let $\mathscr {X}$ be a Banach space, $Z$ a nonempty set, and $u: Z \to \mathscr {X}$. If there are positive constants $M_1$ and $M_2$ such that (i) $\left\| u(z)\right\| \le M_1$ for all $z$ in $Z$ and (ii) for every $x^*$ in $\mathscr {X}$, $\sup \left\{ \left|\langle u(z), x^* \rangle \right|: z \in Z\right\} \ge M_2 \left\| x^*\right\|$; then for every $x$ in $\mathscr {X}$ there is an $f$ in $l^1(Z)$ such that $(*)\ x = \sum \left\{ f(z)u(z): z \in Z\right\}$ and $M_2 \inf \left\| f\right\|_1 \le \left\| x\right\| \le M_1 \inf \left\| f\right\|_1$, where the infimum is taken over all $f$ in $l^1(Z)$ such that $(*)$ holds.

(Bonsall [1986]) Let $m$ be normalized Lebesgue measure on $\partial \mathbb {D}$ and for $\left|z\right| < 1$ and $\left|w\right| = 1$ let $p_z(w) = (1 - \left|z\right|^2)/\left|1 - \bar{z}w\right|^2$. So $p_z$ is the Poisson kernel. Show that if $f \in L^1(m)$, then there is a sequence $\left\{ z_n\right\} \subseteq \mathbb {D}$ and a sequence $\left\{ \lambda_n\right\}$ in $l^1$ such that $(*)\ f = \sum_{n=1}^{\infty } \lambda_n p_{z_n}$. Moreover, $\left\| f\right\|_1 = \inf \sum_{n=1}^{\infty } \left|\lambda_n\right|$, where the infimum is taken over all $\left\{ \lambda_n\right\}$ in $l^1$ such that $(*)$ holds.

If $\mathscr {X}$ is a Banach space and $\mathscr {M}$ and $\mathscr {N}$ are closed subspaces, show that the following statements are equivalent.

Let $L^{1}$ be the usual space of integrable functions on the closed unit interval $J$, relative to Lebesgue measure. Suppose $T \in \mathscr {B}\left(L^{1}, Y\right)$, so that $T^{*} \in \mathscr {B}\left(Y^{*}, L^{\infty }\right)$. Suppose $\mathscr {R}\left(T^{*}\right)$ contains every continuous function on $J$. What can you deduce about $T$?

Suppose $T \in \mathscr {B}(X, Y)$ and $\mathscr {R}(T)$ is closed in $Y$. Prove that

This generalizes the assertions $\alpha = \beta^{*}$ and $\alpha^{*} = \beta$ of Theorem 4.25.

If $T \in \mathscr {B}(X, Y)$ and $\mathscr {R}\left(T^{*}\right) = \mathscr {N}(T)^{\perp }$, prove that $\mathscr {R}(T)$ is closed.