P-Sequence Space admits Schauder Basis

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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:

\(\ds \forall n \in \N : n > N: \, \) \(\ds \norm {\mathbf x - \mathbf s_n}_p^p\) \(=\) \(\ds \sum_{k \mathop = n + 1}^\infty \size {x_k}^p\) Definition of P-Norm
\(\ds \) \(\le\) \(\ds \sum_{k \mathop = N + 1}^\infty \size {x_k}^p\)
\(\ds \) \(<\) \(\ds \epsilon'\)

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, $\ds \mathbf x = \sum_{n = 1}^\infty x_n \mathbf e_n$.

$\Box$


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:

\(\ds \map {\phi_n} {\mathbf z}\) \(=\) \(\ds z_n\)
\(\ds \) \(=\) \(\ds x_n + \lambda y_n\)
\(\ds \) \(=\) \(\ds \map {\phi_n} {\mathbf x} + \lambda \map {\phi_n} {\mathbf y}\)

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:

\(\ds \forall \mathbf x \in \ell^p: \, \) \(\ds \size {\map {\phi_n} {\mathbf x} }\) \(=\) \(\ds \size {x_n}\)
\(\ds \) \(=\) \(\ds \paren {\size{x_n}^p}^{1/p}\)
\(\ds \) \(\le\) \(\ds \sum_{k \mathop = 1}^\infty \paren{\size {x_k}^p}^{\frac 1 p}\)
\(\ds \) \(=\) \(\ds \norm {\mathbf x}_p\) Definition of P-Norm

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:

$\ds \map {\phi_k}{\sum_{i \mathop = 1}^\infty x_i \mathbf e_i} = \sum_{i \mathop = 1}^\infty x_i \map {\phi_k}{\mathbf e_i}$

$\Box$


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:

\(\ds \forall n \in \N: \, \) \(\ds \xi_n\) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \xi_k \delta_{kn}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \xi_k \map {\phi_n} {\mathbf e_k}\)
\(\ds \) \(=\) \(\ds \map {\phi_n} {\sum_{k \mathop = 1}^\infty \xi_k \mathbf e_k}\)
\(\ds \) \(=\) \(\ds \map {\phi_n} {\mathbf x}\)
\(\ds \) \(=\) \(\ds \map {\phi_n} {\sum_{k \mathop = 1}^\infty \xi_k' \mathbf e_k}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \xi_k' \map {\phi_n} {\mathbf e_k}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \xi_k' \delta_{kn}\)
\(\ds \) \(=\) \(\ds \xi_n'\)

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 $\ell^p$.

$\blacksquare$


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