Theorem: Closed Linear Span of Normalized Block Basic Sequence of Schauder Basis in P-Sequence Space is Range of Contractive Projection

Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$. 
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space. 
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis given by P-Sequence Space admits Schauder Basis.
Let $\sequence {y_k}_{k \mathop \in \N}$ be a block basic sequence of $\sequence {e_n}_{n \mathop \in \N}$ with:
:$\norm {y_k}_p = 1$ for each $k \in \N$. 
Let $\sqbrk {y_k}_{k \mathop \in \N}$ be the closed linear span of $\sequence {y_k}_{k \mathop \in \N}$.
Then there exists a bounded projection $Q : \map {\ell_p} \GF \to \sqbrk {y_k}_{k \mathop \in \N}$ such that:
:$\norm {Q x}_p \le \norm x_p$ for each $x \in \map {\ell_p} \GF$. 
In particular, $\sqbrk {y_k}_{k \mathop \in \N}$ is topologically complemented in $\map {\ell_p} \GF$.

Proof:
Let $\sequence {r_j}_{j \mathop \in \N}$ be a strictly increasing sequence in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a sequence in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
We then have:
:$\ds 1 = \norm {y_k}_p = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {a_j}^p$
Define $\sequence {b_j}_{j \mathop \in \N}$ by:
:$\ds b_j = \begin{cases}0 & a_j = 0 \\ \frac {\cmod {a_j}^p} {a_j} & a_j \ne 0\end{cases}$
Let $q \in \hointl 1 \infty$ be such that:
:$\ds \frac 1 p + \frac 1 q = 1$
If $p > 1$, then $q < \infty$ and we have:
{{begin-eqn}}
{{eqn	| l = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {b_j}^q
		| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {a_j}^{q \paren {p - 1} }
}}
{{eqn	| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {a_j}^p 
}}
{{eqn	| r = 1
}}
{{end-eqn}}
If $p = 1$, then $q = \infty$ and we have:
:$\ds \sup_{r_{k - 1} + 1 \le j \le r_k} \cmod {b_j} = \sup_{r_{k - 1} \le j \le r_k} \cmod {\frac {\cmod {a_j} } {a_j} } = 1$ 
In both cases we have:
{{begin-eqn}}
{{eqn	| l = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j a_j
		| r =  \sum_{j \mathop = r_{k - 1} + 1, \, a_j \ne 0}^{r_k} \frac {\cmod {a_j}^p} {a_j} a_j
}}
{{eqn	| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {a_j}^p 
}}
{{eqn	| r = 1
}}
{{end-eqn}}
From Block Basic Sequence is Basic Sequence, $\sequence {y_k}_{k \mathop \in \N}$ is a Schauder basis for $\sqbrk {y_k}_{k \mathop \in \N}$.
Let $\sequence {y_k^\ast}_{k \mathop \in \N}$ be the coordinate functionals associated with $\sequence {y_k}_{k \mathop \in \N}$.
Let $\sequence {e_n^\ast}_{k \mathop \in \N}$ be the coordinate functionals associated with $\sequence {e_n}_{n \mathop \in \N}$.
We aim to show that:
:$\ds y_k^\ast = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j e_j^\ast$
Let:
:$\ds p_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j e_j^\ast$
By Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals, it is enough to show that each $p_k$ is a bounded linear functional, and:
:$\ds \map {p_k} {y_\ell} = \begin{cases}1 & k = \ell \\ 0 & \text{otherwise}\end{cases} = \delta_{k \ell}$
where $\delta_{k \ell}$ denotes the Kronecker $\delta$. 
From Normed Dual Space is Banach Space, we immediately have that each $p_k$ is a bounded linear functional.
{{begin-eqn}}
{{eqn	| l = \map {p_k} {y_\ell}
		| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j \map {e_j^\ast} {y_\ell}
}}
{{eqn	| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j \map {e_j^\ast} {\sum_{i \mathop = r_{\ell - 1} + 1}^{r_\ell} a_i e_i}
}}
{{eqn	| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \sum_{i \mathop = r_{\ell - 1} + 1}^{r_\ell} b_j a_i \delta_{i j}
		| c = {{Defof|Coordinate Functionals Associated with Schauder Basis}}
}}
{{end-eqn}}
Note that the ranges $r_{k - 1} + 1 \le j \le r_k$ are disjoint for distinct $k$. 
Hence $\delta_{i j}$ for $r_{k - 1} + 1 \le j \le r_k$ and $r_{\ell - 1} + 1 \le j \le r_\ell$ if $k \ne \ell$. 
Hence, we have:
:$\ds \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \sum_{i \mathop = r_{\ell - 1} + 1}^{r_\ell} b_j a_i \delta_{i j} = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j a_j = 1$
if $k = \ell$, and:
:$\ds \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \sum_{i \mathop = r_{\ell - 1} + 1}^{r_\ell} b_j a_i \delta_{i j} = 0$
otherwise.
Hence by Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals, we obtain $p_k = y_k^\ast$.
Let $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ be the space of almost-zero sequences.
Define $P : \map {c_{00} } \GF \to \map {\ell_p} \GF$ by:
:$\ds \map P \xi = \sum_{k \mathop = 1}^\infty \map {y_k^\ast} \xi y_k$
Let $\xi_1, \xi_2 \in \map {c_{00} } \GF$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn	| l = \map P {\xi_1 + \lambda \xi_2}
		| r = \sum_{k \mathop = 1}^\infty \map {y_k^\ast} {\xi_1 + \lambda \xi_2} y_k
}}
{{eqn	| r = \sum_{k \mathop = 1}^\infty \paren {\map {y_k^\ast} {\xi_1} + \lambda \map {y_k} {\xi_2} } y_k
		| c = {{Defof|Linear Functional}}
}}
{{eqn	| r = \sum_{k \mathop = 1}^\infty \map {y_k^\ast} {\xi_1} y_k + \lambda \sum_{j \mathop = 1}^\infty \map {y_k^\ast} {\xi_2} y_k
}}
{{eqn	| r = P \xi_1 + \lambda P \xi_2
}}
{{end-eqn}}
Hence $P$ is a linear transformation.
Further, we have:
{{begin-eqn}}
{{eqn	| l = P^2 \xi 
		| r = \sum_{k \mathop = 1}^\infty \map {y_k^\ast} {\sum_{j \mathop = 1}^\infty \map {y_j^\ast} \xi y_j} y_k 
}}
{{eqn	| r = \sum_{k \mathop = 1}^\infty \paren {\sum_{j \mathop = 1}^\infty \map {y_j^\ast} \xi \map {y_k^\ast} {y_j} } y_k
}}
{{eqn	| r = \sum_{k \mathop = 1}^\infty \paren {\sum_{j \mathop = 1}^\infty \map {y_j^\ast} \xi \delta_{k j} } y_k 
}}
{{eqn	| r = \sum_{k \mathop = 1}^\infty \map {y_k^\ast} \xi y_k 
}}
{{eqn	| r = P \xi 
}}
{{end-eqn}}
So $P$ is a projection.
Finally, since $\map {y_k^\ast} {y_j} = \delta_{k j}$, we have that:
:$P y_j = y_j$ for each $j \in \N$
and further:
:$P \sqbrk {\map {c_{00} } \GF} \subseteq \map \span {y_j : j \in \N} \subseteq \sqbrk {y_k}_{k \mathop \in \N}$
We aim to extend $P$ to the whole of $\map {\ell_p} \GF$.
We now show that:
:$\norm {P \xi}_p \le \norm \xi_p$ for each $x \in \map {\ell_p} \GF$.
For each $\xi \in \map {c_{00} } \GF$, we have:
:$\ds \cmod {\map {y_k^\ast} \xi} = \cmod {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j \map \xi j}$
In the case $p > 1$, we have:
{{begin-eqn}}
{{eqn	| l = \cmod {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j \map \xi j}
		| o = \le
		| r = \paren {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {b_j}^q}^{1/q} \paren {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j}^p}^{1/p}
		| c = Hölder's Inequality for Sums: Finite
}}
{{eqn	| r = \paren {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j}^p}^{1/p}
}}
{{end-eqn}}
In the case $p = 1$, we have:
{{begin-eqn}}
{{eqn	| l = \cmod {\sum_{j \mathop = r_{k - 1} + 1}^{r_k} b_j \map \xi j}
		| o = \le
		| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {b_j \map \xi j}
		| c = Triangle Inequality: Complex Numbers
}}
{{eqn	| o = \le
		| r = \sup_{r_{k - 1} + 1 \le j \le r_k} \cmod {b_j} \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j} 
}}
{{eqn	| r = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j} 
}}
{{end-eqn}}
Hence we have:
:$\ds \cmod {\map {y_k^\ast} \xi}^p \le \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j}^p$
Hence:
{{begin-eqn}}
{{eqn	| l = \norm {P \xi}_p
		| r = \paren {\sum_{k \mathop = 1}^\infty \cmod {\map {y_k^\ast} \xi}^p}^{1/p}
}}
{{eqn	| o = \le
		| r = \paren {\sum_{k \mathop = 1}^\infty \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod {\map \xi j}^p}^{1/p}
}}
{{eqn	| r = \norm \xi_p 
}}
{{end-eqn}}
Hence $P : \map {c_{00} } \GF \to \sqbrk {y_k}_{k \mathop \in \N}$ is bounded.
From Closed Subspace of Banach Space forms Banach Space, $\struct {\sqbrk {y_k}_{k \mathop \in \N}, \norm {\, \cdot \,}_p}$ is a Banach space.
Hence from Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain, $P$ extends to a bounded linear transformation $Q : \map {\ell_p} \GF \to \sqbrk {y_k}_{k \mathop \in \N}$ with:
:$\norm {Q x}_p \le \norm x_p$ for each $x \in \map {\ell_p} \GF$.
We finish by verifying that $Q^2 = Q$ and $Q \sqbrk {\map {\ell_p} \GF} = \sqbrk {y_k}_{k \mathop \in \N}$.
Let $x \in \map {\ell_p} \GF$.
From Space of Almost-Zero Sequences is Everywhere Dense in P-Sequence Space, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\map {c_{00} } \GF$ such that:
:$x_n \to x$ in $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$. 
We have:
:$\map Q {Q x_n} = x_n$ for each $n \in \N$.
Since $Q$ is bounded, we have $Q x_n \to Q x$.
Applying this idea again, we also have $Q x_n = Q^2 x_n \to Q^2 x$.
Hence $Q^2 x = Q x$ by Convergent Sequence in Normed Vector Space has Unique Limit.
Since $x \in \map {\ell_p} \GF$ was arbitrary, we have $Q^2 = Q$.
Now take $u \in \sqbrk {y_k}_{k \mathop \in \N}$. 
Write:
:$\ds u = \sum_{j \mathop = 1}^\infty \map {y_k^\ast} u y_k$
Write:
:$\ds u = \sum_{j \mathop = 1}^n \map {y_k^\ast} u y_k$ 
Since $P y_k = Q y_k = y_k$ for each $k \in \N$, we obtain:
{{begin-eqn}}
{{eqn	| l = Q u_n
		| r = \map P {\sum_{j \mathop = 1}^n \map {y_k^\ast} u y_k}
}}
{{eqn	| r = \sum_{j \mathop = 1}^n \map {y_k^\ast} u P y_k 
		| c = $P$ is linear
}}
{{eqn	| r = \sum_{j \mathop = 1}^n \map {y_k^\ast} u y_k 
}}
{{eqn	| r = u_n 
}}
{{end-eqn}}
We have $u_n \to u$, and hence $P u_n \to P u$.
Hence we have:
:$Q u = u$ for each $u \in \sqbrk {y_k}_{k \mathop \in \N}$.
Since $Q \sqbrk {\map {\ell_p} \GF} \subseteq \sqbrk {y_k}_{k \mathop \in \N}$, we must have:
:$Q \sqbrk {\map {\ell_p} \GF} = \sqbrk {y_k}_{k \mathop \in \N}$
By the definition of a topologically complemented subspace, we have that:
:$\sqbrk {y_k}_{k \mathop \in \N}$ is topologically complemented in $\map {\ell_p} \GF$ {{iff}} there exists a bounded projection $Q : \map {\ell_p} \GF \to \sqbrk {y_k}_{k \mathop \in \N}$.
We have found such a $Q$, and so we are done.
{{qed}}