Theorem: Normed Dual Space of 1-Sequence Space is Isometrically Isomorphic to Space of Bounded Sequences

Let $\GF \in \set {\R, \C}$.
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 $\struct {\map {\ell_1^\ast} \GF, \norm {\, \cdot \,}_{\ell_1^\ast} }$ 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:
:$\ds \map {f_x} y = \sum_{i \mathop = 1}^\infty x_i y_i$
Then $f_x \in \map {\ell_1^\ast} \GF$ for each $x \in \map {\ell_\infty} \GF$ and further $T_1 : \map {\ell_\infty} \GF \to \map {\ell_1^\ast} \GF$ defined by $\map {T_1} x = f_x$ for each $x \in \map {\ell_\infty} \GF$ is an isometric isomorphism.
Hence, $\struct {\map {\ell_1^\ast} \GF, \norm {\, \cdot \,}_{\ell_1^\ast} }$ is isometrically isomorphic to $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$.

Proof:
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {\ell_1} \GF$ given by P-Sequence Space admits Schauder Basis.
We first show that $T_1$ is a linear isometry.
We first argue that $f_x$ is well-defined for each $x \in \map {\ell_\infty} \GF$.
We have $\cmod {x_i} \le \norm x_\infty$ for each $i \in \N$ and hence:
:$\ds \sum_{i \mathop = 1}^\infty \cmod {x_i y_i} \le \norm x_\infty \sum_{i \mathop = 1}^\infty \cmod {y_i} = \norm x_\infty \norm y_1 < \infty$
Hence from Absolutely Convergent Series is Convergent, we have that:
:$\ds \sum_{i \mathop = 1}^\infty x_i y_i$ is convergent.
Now we argue that $f_x$ is a linear functional. 
Let $y, z \in \map {\ell_1} \GF$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn	| l = \map {f_x} {y + \lambda z}
		| r = \sum_{i \mathop = 1}^\infty x_i \paren {y_i + \lambda z_i}
}}
{{eqn	| r = \sum_{i \mathop = 1}^\infty x_i y_i + \lambda \sum_{i \mathop = 1}^\infty x_i z_i
		| c = Linear Combination of Convergent Series
}}
{{eqn	| r = \map {f_x} y + \lambda \map {f_x} z
}}
{{end-eqn}}
Hence $f_x$ is linear functional. 
We now show that $f_x$ is a bounded linear functional with:
:$\norm {f_x}_{\ell_1^\ast} = \norm x_\infty$
We firstly have, for $x \in \map {\ell_\infty} \GF$ and $y \in \map {\ell_\infty} \GF$:
:$\ds \cmod {\map {f_x} y} = \cmod {\sum_{i \mathop = 1}^\infty x_i y_i} \le \sum_{i \mathop = 1}^\infty \cmod {x_i y_i} \le \norm x_\infty \norm y_1$ 
Hence we have that $f_x$ is a bounded linear functional with:
:$\norm {f_x}_{\ell_1^\ast} \le \norm x_\infty$
It is now enough to show that:
:$\norm {f_x}_{\ell_1^\ast} \ge \norm x_\infty$ 
Let $0 < \epsilon < \norm x_\infty$.
By the definition of supremum, we can pick $n \in \N$ such that:
:$\cmod {x_n} > \norm x_\infty - \epsilon > 0$
Set:
:$\ds y = \frac 1 {x_n} e_n \in \map {\ell_1} \GF$
We then have that:
:$\ds \map {f_x} y = \sum_{i \mathop = 1}^\infty \frac {x_i} {x_n} \delta_{i n} = \frac {x_n} {x_n} = 1$
where $\delta$ denotes the Kronecker $\delta$.
We then have:
{{begin-eqn}}
{{eqn	| l = 1
		| r = \cmod {\map {f_x} y}
}}
{{eqn	| o = \le
		| r = \norm {f_x}_{\ell_1^\ast} \norm y_1
		| c = Fundamental Property of Norm on Bounded Linear Functional
}}
{{eqn	| r = \norm {f_x}_{\ell_1^\ast} \frac 1 {\cmod {x_n} } 
}}
{{eqn	| o = <
		| r = \norm {f_x}_{\ell_1^\ast} \frac 1 {\norm x_\infty - \epsilon}
}}
{{end-eqn}}
Hence we have:
:$\norm x_\infty - \epsilon < \norm {f_x}_{\ell_1^\ast}$ for each $\epsilon > 0$.
Hence we have $\norm {f_x}_{\ell_1^\ast} = \norm x_\infty$.
We now show that $T_1$ is linear.
Let $x, y \in \map {\ell_\infty} \GF$, $z \in \map {\ell_1} \GF$ and $\lambda \in \GF$.
We then have:
{{begin-eqn}}
{{eqn	| l = \map {f_{x + \lambda y} } z
		| r = \sum_{i \mathop = 1}^\infty \paren {x_i + \lambda y_i} z_i
}}
{{eqn	| r = \sum_{i \mathop = 1}^\infty x_i z_i + \lambda \sum_{i \mathop = 1}^\infty y_i z_i 
}}
{{eqn	| r = \map {f_x} z + \lambda \map {f_y} z 
}}
{{end-eqn}}
that is:
:$f_{x + \lambda y} = f_x + \lambda f_y$
So that:
:$\map {T_1} {x + \lambda y} = \map {T_1} x + \lambda \map {T_1} y$
We finally show that $T_1$ is surjective.
Let $f \in \map {\ell_1^\ast} \GF$.
Define $x : \N \to \GF$ by:
:$x_i = \map f {e_i}$ for each $i \in \N$.
We show that $x \in \map {\ell_\infty} \GF$.
From Fundamental Property of Norm on Bounded Linear Functional, we have:
:$\cmod {x_i} = \map f {e_i} \le \norm f_{\ell_1^\ast} \norm {e_i}_1 = \norm f_{\ell_1^\ast}$
That is, $\norm x_\infty \le \norm f_{\ell_1^\ast} < \infty$ and hence $x \in \map {\ell_\infty} \GF$.
We now show that $f = f_x$.
For each $i \in \N$, we have:
:$\map {f_x} {e_i} = \map f {e_i}$
For each $\lambda_1, \ldots, \lambda_n \in \GF$, we have:
:$\ds \map {f_x} {\sum_{i \mathop = 1}^n \lambda_i e_i} = \map f {\sum_{i \mathop = 1}^n \lambda_i e_i}$
Hence:
:$\map {f_x} y = \map f y$ for each $y \in \map \span {\set {e_n : n \in \N} }$.
From P-Sequence Space admits Schauder Basis, $\set {e_n : n \in \N}$ is a Schauder basis for $\map {\ell_1} \GF$.
Hence:
:$\ds \map {\ell_1} \GF = \map \cl {\map \span {\set {e_n : n \in \N} } }$
From Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide, we therefore conclude that:
:$\map {f_x} y = \map f y$ for each $y \in \map {\ell_1} \GF$.
Hence $f = f_x$. 
That is, for each $f \in \map {\ell_1^\ast} \GF$ there exists $x \in \map {\ell_\infty} \GF$ with $f = f_x$. 
Hence $T_1$ is surjective.
We have established that $T_1$ is a surjective linear isometry.
From Linear Isometry is Injective: Corollary, $T_1$ is an isometric isomorphism.
This completes the proof.
{{qed}}
Category:P-Sequence Spaces
Category:Space of Bounded Sequences
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