Theorem: Unconditional Weak-* Convergence of Sum of Schauder Basis of Zero-Limit Sequences in Space of Bounded Sequences

Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {c_0} \GF$ given by Space of Zero-Limit Sequences admits Schauder Basis.
Let $\struct {\map {\ell_1^\ast} \GF, \norm {\, \cdot \,}_{\ell_1^\ast} }$ be the normed dual space of $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$.
For each $x \in \map {\ell_\infty} \GF$, define $f_x : \map {\ell_1} \GF \to \GF$ by:
:$\ds \map {f_x} y = \sum_{i \mathop = 1}^\infty x_i y_i$
By Normed Dual Space of 1-Sequence Space is Isometrically Isomorphic to Space of Bounded Sequences, we have that $f_x \in \map {\ell_1^\ast} \GF$ for each $x \in \map {\ell_\infty} \GF$.
Define $T_1 : \struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty} \to \struct {\map {\ell_1^\ast} \GF, \norm {\, \cdot \,}_{\ell_1^\ast} }$ by: 
:$\map {T_1} x = f_x$ for each $x \in \map {\ell_\infty} \GF$.
Let $w^\ast$ be the weak-$\ast$ topology on $\map {\ell_1^\ast} \GF$.
Consider the mapping $T_1^{-1} : \struct {\map {\ell_1^\ast} \GF, w^\ast} \to \map {\ell_\infty} \GF$.
Let $w^\ast_{\ell_\infty}$ be the topology induced by $T_1^{-1}$ considered this way. 
Define $\mathbf 1 : \N \to \GF$ by:
:$\map {\mathbf 1} n = 1$ for each $n \in \N$. 
Then:
:$\ds \sum_{n \mathop = 1}^\infty e_n$ converges unconditionally to $\mathbf 1$ in $\struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} }$.

Proof:
From Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology:
:$T_1 : \struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} } \to \struct {\map {\ell_1^\ast} \GF, w^\ast}$ is a homeomorphism.
Hence it suffices to show that:
:$\ds \sum_{n \mathop = 1}^\infty f_{e_n}$ converges unconditionally to $f_{\mathbf 1}$ in $\struct {\map {\ell_1^\ast} \GF, w^\ast}$.
Let $x \in \map {\ell_1} \GF$. 
Let $\pi : \N \to \N$ be a permutation. 
From Manipulation of Absolutely Convergent Series: Permutation:
:$\ds \sum_{n \mathop = 1}^\infty x_{\map \pi n}$ converges absolutely. 
We have:
{{begin-eqn}}
{{eqn	| l = \sum_{n \mathop = 1}^N \map {f_{e_{\map \pi n} } } x
		| r = \sum_{n \mathop = 1}^N \sum_{j \mathop = 1}^\infty \delta_{j \map \pi n} x_j
}}
{{eqn	| r = \sum_{n \mathop = 1}^N x_{\map \pi n}
}}
{{eqn	| o = \to
		| r = \sum_{n \mathop = 1}^\infty x_{\map \pi n}
		| c = as $N \to \infty$,
}}
{{eqn	| r = \sum_{n \mathop = 1}^\infty x_n
		| c = Manipulation of Absolutely Convergent Series: Permutation
}}
{{eqn	| r = \sum_{n \mathop = 1}^\infty \map {\mathbf 1} n x_n
}}
{{eqn	| r = \map {f_{\mathbf 1} } x
}}
{{end-eqn}}
Hence:
:$\ds \sum_{n \mathop = 1}^\infty f_{e_n}$ converges unconditionally to $f_{\mathbf 1}$ in $\struct {\map {\ell_1^\ast} \GF, w^\ast}$.
That is:
:$\ds \sum_{n \mathop = 1}^\infty \map {T_1} {e_n}$ converges unconditionally to $\map {T_1} {\mathbf 1}$ in $\struct {\map {\ell_1^\ast} \GF, w^\ast}$.
Since $T_1 : \struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} } \to \struct {\map {\ell_1^\ast} \GF, w^\ast}$ is a linear homeomorphism, we have that:
:$\ds \sum_{n \mathop = 1}^\infty e_n$ converges unconditionally to $\mathbf 1$ in $\struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} }$
{{qed}}
Category:Unconditionally Convergent Series
Category:Space of Bounded Sequences
Category:Weak-* Topologies
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