Theorem: P-Sequence Space is Reflexive for Finite p Greater Than One

Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Then $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is reflexive.

Proof:
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 $\struct {\map {\ell_q^\ast} \GF, \norm {\, \cdot \,}_{\ell_q^\ast} }$ be the normed dual space of $\struct {\map {\ell_q} \GF, \norm {\, \cdot \,}_q}$.
Let $\struct {\map {\ell_p^{\ast \ast} } \GF, \norm {\, \cdot \,}_{\ell_p^{\ast \ast} } }$ be the second normed dual of $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $\struct {\map {\ell_q^{\ast \ast} } \GF, \norm {\, \cdot \,}_{\ell_q^{\ast \ast} } }$ be the second normed dual of $\struct {\map {\ell_q} \GF, \norm {\, \cdot \,}_q}$.
Let $\iota : \map {\ell_p} \GF \to \map {\ell_p^{\ast \ast} } \GF$ be the evaluation linear transformation.
Let $F \in \map {\ell_p^{\ast \ast} } \GF$.
We aim to find $z \in \map {\ell_p} \GF$ such that $F = \iota z$.
Let:
:$\ds q = \frac p {p - 1} \in \openint 1 \infty$
so that:
:$\ds \frac 1 p + \frac 1 q = 1$
For each $x \in \map {\ell_q} \GF$, define $f_x : \map {\ell_p} \GF \to \GF$ by:
:$\ds \map {f_x} y = \sum_{i \mathop = 1}^\infty x_i y_i$ for each $y \in \map {\ell_p} \GF$.
This is well-defined from Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space.
From Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space, we can define an isometric isomorphism $T_p : \map {\ell_q} \GF \to \map {\ell_p^\ast} \GF$ by:
:$\map {T_p} x = f_x$ for each $x \in \map {\ell_q} \GF$.
For each $u \in \map {\ell_p} \GF$, define $g_u : \map {\ell_q} \GF \to \GF$ by:
:$\ds \map {f_u} z = \sum_{i \mathop = 1}^\infty u_i z_i$ for each $z \in \map {\ell_q} \GF$.
From Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space, we can define an isometric isomorphism $T_q : \map {\ell_p} \GF \to \map {\ell_q^\ast} \GF$ by:
:$\map {T_p} u = g_u$ for each $x \in \map {\ell_p} \GF$.
Since $F : \map {\ell_p^\ast} \GF \to \GF$ and $T_p : \map {\ell_q} \GF \to \map {\ell_p^\ast} \GF$ are bounded linear transformations, we have that:
:$F \circ T_p : \map {\ell_q} \GF \to \GF$ is a bounded linear functional.
Since $T_q : \map {\ell_p} \GF \to \map {\ell_q^\ast} \GF$ is surjective by Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space, there exists $z \in \map {\ell_q} \GF$ such that:
:$F \circ T_p = \map {T_q} z$
We now want to show that:
:$\map F f = \map {\paren {\iota z} } f = \map f z$ for each $f \in \map {\ell_p^\ast} \GF$.
Since $T_p$ is surjective, it is enough to show that:
:$\map F {f_x} = \map {f_x} z$ for each $x \in \map {\ell_q} \GF$.
Indeed we have:
{{begin-eqn}}
{{eqn	| l = \map F {f_x}
		| r = \map {\paren {F \circ T_p} } x
}}
{{eqn	| r = \map {\paren {\map {T_q} z} } x
}}
{{eqn	| r = \sum_{j \mathop = 1}^\infty x_i z_i
}}
{{eqn	| r = \map {f_x} z 
}}
{{end-eqn}}
Hence for all $F \in \map {\ell_p^{\ast \ast} } \GF$, there exists $z \in \map {\ell_p} \GF$ such that $F = \iota z$.
Hence $\iota : \map {\ell_p} \GF \to \map {\ell_p^{\ast \ast} } \GF$ is a surjection.
Hence from Normed Vector Space is Reflexive iff Surjective Evaluation Linear Transformation, we have that $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is reflexive.
{{qed}}
Category:Reflexive Spaces
Category:P-Sequence Spaces
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