Theorem: Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space

Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let:
:$\ds q = \frac p {p - 1}$
so that:
:$\ds \frac 1 p + \frac 1 q = 1$
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the normed dual space of $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $\struct {\map {\ell_q} \GF, \norm {\, \cdot \,}_q}$ be the $q$-sequence space.
For each $x \in \map {\ell_q} \GF$, define $f_x : \map {\ell_p} \GF \to \GF$ by:
:$\ds \map {f_x} y = \sum_{i \mathop = 1}^\infty x_i y_i$
Then $f_x \in \map {\ell_p^\ast} \GF$ for each $x \in \map {\ell_q} \GF$.
Further the mapping $T_p : \map {\ell_q} \GF \to \map {\ell_p^\ast} \GF$ defined by:
:$\map {T_p} x = f_x$ for each $x \in \map {\ell_q} \GF$
is an isometric isomorphism.
Hence, $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ is isometrically isomorphic to $\struct {\map {\ell_q} \GF, \norm {\, \cdot \,}_q}$.

Proof:
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {\ell_p} \GF$ given by P-Sequence Space admits Schauder Basis.
We first show that $T_p$ is a linear isometry.
Firstly, $f_x$ is well-defined by Hölder's inequality.
Now we argue that $f_x$ is a linear functional. 
Let $y, z \in \map {\ell_p} \GF$ 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:
:$\norm {f_x}_{\ell_p^\ast} = \norm x_q$
Firstly, by Hölder's inequality we have:
:$\ds \cmod {\map {f_x} y} = \cmod {\sum_{i \mathop = 1}^\infty x_i y_i} \le \sum_{i \mathop = 1}^\infty \cmod {x_i y_i} \le \norm x_q \norm y_p$
Hence $f_x$ is bounded with:
:$\norm {f_x}_{\ell_p^\ast} \le \norm x_q$ 
Define $y : \N \to \GF$ by:
:$\ds y_i = \begin{cases}\frac {\cmod {x_i}^q} {x_i} & x_i \ne 0 \\ 0 & x_i = 0\end{cases}$
Since:
:$\ds \frac 1 p + \frac 1 q = 1$
we have:
:$p q = p + q$
and hence:
:$q = p \paren {q - 1}$
We therefore have:
{{begin-eqn}}
{{eqn	| l = \norm y_p^p
		| r = \sum_{i \mathop = 1}^\infty \cmod {y_i}^p
}}
{{eqn	| r = \sum_{i \mathop = 1}^\infty \cmod {x_i}^{p \paren {q - 1} }
}}
{{eqn	| r = \sum_{i \mathop = 1}^\infty \cmod {x_i}^q 
}}
{{eqn	| r = \norm x_q^q
}}
{{eqn	| o = <
		| r = \infty 
		| c = {{Defof|p-Sequence Space}}
}}
{{end-eqn}}
Hence $y \in \map {\ell_p} \GF$.
Since $q = p \paren {q - 1}$, we further have:
:$\dfrac q p = q - 1$
We now have:
:$\norm y_p = \norm x_q^{q/p} = \norm x_q^{q - 1}$
and hence:
{{begin-eqn}}
{{eqn	| l = \map {f_x} y
		| r = \sum_{i \mathop = 1}^\infty x_i y_i
}}
{{eqn	| r = \sum_{i \mathop = 1}^\infty \cmod {x_i}^q
}}
{{eqn	| r = \norm x_q^q
}}
{{eqn	| r = \norm x_q \norm x_q^{q - 1}
}}
{{eqn	| r = \norm x_q \norm y_p
}}
{{end-eqn}}
Hence we have:
:$\norm {f_x}_{\ell_p^\ast} \ge \norm x_q$
Hence we have:
:$\norm {f_x}_{\ell_p^\ast} = \norm x_q$
We now show that $T_1$ is linear.
Let $x, y \in \map {\ell_q} \GF$, $z \in \map {\ell_p} \GF$ 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.
We can firstly see that ${\mathbf 0}_{\ell_p^\ast} = f_{ {\mathbf 0}_{\ell_p} }$. 
Let $f \in \map {\ell_p^\ast} \GF \setminus \set { {\mathbf 0}_{\ell_p^\ast} }$.
Define $x : \N \to \GF$ by:
:$x_i = \map f {e_i}$ for each $i \in \N$.
Since $f \ne {\mathbf 0}_{\ell_p^\ast}$, we have $x_i \ne 0$ for some $i \in \N$.
We first show that $x \in \map {\ell_q} \GF$.
For each $N \in \N$, we define $\phi_N \in \map {\ell_p} \GF$ by:
:$\ds \map {\phi_N} n = \begin{cases}\frac {\cmod {x_n}^q} {x_n} & x_n \ne 0, \, n \le N \\ 0 & \text{otherwise}\end{cases}$
that is:
:$\ds \phi_N = \sum_{n \mathop = 1}^N \frac {\cmod {x_n}^q} {x_n} e_n$ 
As before, we have:
{{begin-eqn}}
{{eqn	| l = \norm {\phi_N}_p^p
		| r = \sum_{i \mathop = 1}^N \cmod {\map {\phi_N} i}^p
}}
{{eqn	| r = \sum_{i \mathop = 1}^N \cmod {x_i}^{p \paren {q - 1} }
}}
{{eqn	| r = \sum_{i \mathop = 1}^N \cmod {x_i}^q 
}}
{{end-eqn}}
Further:
{{begin-eqn}}
{{eqn	| l = \map f {\phi_N}
		| r = \map f {\sum_{n \mathop = 1}^N \frac {\cmod {x_n}^q} {x_n} e_n}
}}
{{eqn	| r = \sum_{n \mathop = 1}^N \frac {\cmod {x_n}^q} {x_n} \map f {e_n}
}}
{{eqn	| r = \sum_{n \mathop = 1}^N \frac {\cmod {x_n}^q} {x_n} x_n
}}
{{eqn	| r = \sum_{n \mathop = 1}^N \cmod {x_n}^q
}}
{{end-eqn}}
Recall that since $x \ne {\mathbf 0}_{\ell_q}$, we have:
:$\ds \sum_{i \mathop = 1}^N \cmod {x_i}^q \ne 0$ 
We have:
{{begin-eqn}}
{{eqn	| l = \sum_{n \mathop = 1}^N \cmod {x_n}^q
		| r = \map f {\phi_N}
}}
{{eqn	| o = \le
		| r = \norm f_{\ell_p^\ast} \norm {\phi_N}_p
		| c = Fundamental Property of Norm on Bounded Linear Functional
}}
{{eqn	| r = \norm f_{\ell_p^\ast} \paren {\sum_{i \mathop = 1}^N \cmod {x_i}^q}^{1/p}
}}
{{end-eqn}}
Rearranging:
:$\ds \paren {\sum_{i \mathop = 1}^N \cmod {x_i}^q}^{1 - 1/p} \le \norm f_{\ell_p^\ast}$
Since:
:$\ds 1 - \frac 1 p = \frac 1 q$
we have that:
:$\norm x_{\ell_q} \le \norm f_{\ell_p^\ast} < \infty$
Hence $x \in \map {\ell_q} \GF$.
We verify 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 $\map {\ell_p} \GF$.
Hence:
:$\ds \map {\ell_p} \GF = \map \cl {\map \span {\set {e_n : n \in \N} } }$
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 $y \in \map {\ell_p} \GF$.
Hence $f = f_x$. 
That is, for each $f \in \map {\ell_p^\ast} \GF$ there exists $x \in \map {\ell_q} \GF$ with $f = f_x$. 
Hence $T_p$ is surjective.
We have established that $T_p$ is a surjective linear isometry.
From Linear Isometry is Injective: Corollary, $T_p$ is an isometric isomorphism.
This completes the proof.
{{qed}}
Category:P-Sequence Spaces
Category:Normed Dual Spaces
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