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 defnition, $\phi_n$ 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 Transformations on Normed Vector Space, $\phi_n$ 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 $\ell^p$.