P-Sequence Space admits Schauder Basis

== Theorem ==
Let $1 \le p < \infty$.
Let $\ell^p$ be the $p$-sequence space.
Let $\sequence {\mathbf e_n}_{n \mathop \in \N } \in \ell^p$ be a sequence such that:
:$\mathbf e_n = \tuple {\underbrace{0, \ldots, 0}_n, 1, 0, \ldots}$
Then $\set {\mathbf e_n : n \in \N}$ is a Schauder basis for $\ell^p$.
== Proof ==
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} = \tuple {x_1, x_2, x_3, \ldots} \in \ell^p$.
=== $\sequence {x_n}_{n \mathop \in \N}$ converges in $\ell^p$ with basis $\mathbf e_n$ ===
By definition of the $p$-sequence space:
$\ds \sum_{n \mathop = 0}^\infty \size {x_n}^p < \infty$
By Tail of Convergent Series tends to Zero:
:$\ds \forall \epsilon' \in \R_{>0} : \exists N \in \N : \forall n \in \N : n > N \implies \sum_{k \mathop = n}^\infty \size {x_k}^p < \epsilon'$
Let $\ds \mathbf s_n := \sum_{k \mathop = 0}^n x_k \mathbf e_k$.
Then:
:$\ds \forall n \in \N : \mathbf x - \mathbf s_n = \tuple {0, \ldots, 0, x_{n + 1}, x_{n + 2}, x_{n + 3}, \ldots}$
Therefore: 
Let $\epsilon' = \epsilon^p$.
Then:
:$\norm {\mathbf x - \mathbf s_n}_p < \epsilon$.
Altogether:
:$\forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \norm {\mathbf x - \mathbf s_n}_p < \epsilon$
By definition, $\sequence {\mathbf s_n}_{n \in \N}$ converges in $\ell^p$ to $\mathbf x$.
In other words, $\ds \mathbf x = \sum_{n = 1}^\infty x_n \mathbf e_n$.
Let $\phi_n : \ell^p \to \R$ be a map such that:
:$\mathbf x = \tuple {x_1, x_2, x_3, \ldots} \stackrel {\phi_n} {\mapsto} x_n$.
=== $\phi_n$ commutes with infinite sum ===
We have that $\ell^p$ is a vector space.
Let $\mathbf z \in \ell^p$ be such that:
:$\mathbf z = \mathbf x + \lambda \mathbf y$
where $\mathbf x, \mathbf y \in \ell^p, \lambda \in \R$.
Then:
By definition, $\phi_n$ is linear.
Then for $\mathbf x - \mathbf s_k \in \ell^p$ we have:
:$\ds \map {\phi_n} {\mathbf x - \mathbf s_k} = \map {\phi_n}{\mathbf x} - \map {\phi_n}{\mathbf s_k} = \map {\phi_n}{\sum_{i \mathop = 1}^\infty x_i \mathbf e_i} - \sum_{i \mathop = 1}^k x_i \map {\phi_n}{\mathbf e_i}$
Furthermore:
By Continuity of Linear Transformation between Normed Vector Spaces, $\phi_n$ is continuous.
Then:
:$\size {\map {\phi_n} {\mathbf x - \mathbf s_k} } \le \norm {\mathbf x - \mathbf s_k}_p$
Altogether:
:$\ds \forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \size {\map {\phi_k}{\sum_{i \mathop = 1}^\infty x_i \mathbf e_i} - \sum_{i \mathop = 1}^n x_i \map {\phi_k}{\mathbf e_i}} < \epsilon$
Hence:
:$\ds \map {\phi_k}{\sum_{i \mathop = 1}^\infty x_i \mathbf e_i} = \sum_{i \mathop = 1}^\infty x_i \map {\phi_k}{\mathbf e_i}$
=== Uniqueness of expansion coefficients ===
Suppose $\ds \mathbf x = \sum_{k \mathop = 1}^\infty \xi_k \mathbf e_k = \sum_{k \mathop = 1}^\infty \xi_k' \mathbf e_k$.
Then:
Therefore:
:$\forall n \in \N : \xi_n = \xi'_n$
and the expression of $\mathbf x$ in basis $\mathbf e_n$ is unique.
By definition, $\set {\mathbf e_n : n \in \N}$ is a Schauder basis for $\ell^p$.
== Sources ==
* : Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$. Operator norm and the normed space $\map {CL} {X, Y}$
Category:Schauder Bases
Category:P-Sequence Spaces
Category:Bases of Vector Spaces