Schauder decomposition of a Banach space E From book J. T. Marti, Introduction to the Theory of Bases, 2013 and
I. Singer, Bases in Banach Spaces II, 2011.
Theorem
For a Banach space $E$ the following statements are
equivalent:

*

*There is a sequence $\{M_i\}$ of closed linear subspaces of $E$ which is a
Schauder decomposition of $E.$


*There exists a sequence $\{v_n\} \subset L(E, E)$ (i.e. of continuous projections of $E$) with $v_n \neq 0 \ (n = 1,2, ...),$ satisfying $v_i v_j = \delta_{ij} v_i = \delta_{ij} v_j \ (i, j= 1, 2, ...)$ and $x = \displaystyle \sum_{i=1}^\infty v_i(x)$ (i.e. $ x = \displaystyle \lim_n \sum_{i \leq n} v_i x)$ for each $x$ in $E.$


*There exists a sequence $\{v_n\} \subset L(E, E)$ with $v_n \neq 0 (n = 1,2, ...),$
satisfying $v_i v_j = \delta_{ij} v_i = \delta_{ij} v_j (i, j= 1, 2, ...)$, $\big [ \displaystyle \bigcup_{n=1}^\infty v_n(E) \big ] = E$ and $\displaystyle \sup_{1 \leq n < \infty}\|\sum_{i=1}^n v_i\|<\infty$ where $\big[\displaystyle \bigcup_{n=1}^\infty v_n(E) \big]$ is the (closed linear) subspace spanned by $\displaystyle \bigcup_{n=1}^\infty v_n(E).$


*There exists a sequence $\{u_n\} \subset L(E, E)$ (i.e. of continuous projections of $E$) with $u_1 \neq 0, u_n \neq u_{n-1}
(n = 2,3, ...),$ satisfying $u_n u_m = u_m u_n = u_{\min(n,m)} (n, m = 1, 2, ...)$ and $x = \displaystyle \lim_{n \rightarrow \infty} u_n(x)$ for each $x$ in $E.$


*There exists a sequence $\{u_n\} \subset L(E, E)$ with $u_1 \neq 0, u_n \neq u_{n-1}
(n = 2,3, ...),$ satisfying $u_n u_m = u_m u_n = u_{\min(n,m)} (n, m = 1, 2, ...)$ , $\overline{\displaystyle \bigcup_{n=1}^\infty u_n(E)} = E$ and $\displaystyle\sup_{1 \leq n < \infty} \|u_n\| < \infty$ where $\overline{\displaystyle \bigcup_{n=1}^\infty u_n(E)}$ is the closure of $\displaystyle \bigcup_{n=1}^\infty u_n(E)$ in the norm-topology of $E.$
The subspaces $M_i$ and the projections $v_i$ and $u_i$ are related by $v_i(E) = M_i (i = 1,2,...)$
and $u_n = \displaystyle \sum_{i \leq n} v_i, n= 1,2, ....$ (or $v_1 = u_1, v_n = u_n - u_{n-1} (n = 2,3,...)$).
Proposition
Let $\{x_n\}_{n=1}^\infty$ be a sequence of nonzero vectors in $X.$ Then $\{x_n\}_{n \in \mathbb{N}}$ is a Schauder basis of $X$ if and only if

*

*there exists a number $K > 0$ such that, for all integers $p$ and $q,$ with $p \leq q,$ and all scalars $a_0 ... a_q,$ one has:
$$\|\sum_0^p a_n x_n\| \leq K \|\sum_0^q a_n x_n\|.$$

*The closed linear span of $\{x_n\}_{n=1}^\infty$ is all of $X$ i.e. $\{x_n\}$ has dense linear span in $X.$
Thus, a sequence $(x_n)_{n \in \mathbb{N}}$ is basic if and only if condition (1) is satisfied.

In the proof of the theorem in the equivalence of 4 and 5, it is said that as in the last Proposition it is easily seen that it is possible to replace the condition
$x = \displaystyle \lim_{n \rightarrow \infty} u_n(x)$ for each $x$ in $E$ by the apparently weaker conditions $\displaystyle\sup_{1 \leq n < \infty} \|u_n\| < \infty$
and $\overline{\displaystyle \bigcup_{n=1}^\infty u_n(E)} = E.$ This finishes the proof of the theorem.
Q1 I can't understand how this proposition helps in the proof of the equivalence of item 4 and item 5 of the theorem. Any help will be appreciated.
Q2 How to prove that $\big [ \displaystyle \bigcup_{n=1}^\infty v_n(E) \big ] = E$ is equivalent to $\overline{\displaystyle \bigcup_{n=1}^\infty u_n(E)} = E$
 A: As in the last Proposition of the question we have the following theorem from Book
Introduction to the Theory of Bases by J. T. Marti
Theorm
Let $\{M_i\}$ be a sequence of closed subspaces in a Banach space $X$ such that $\overline{sp}\displaystyle\bigcup_{i=1}^\infty M_i = X$ i.e. $sp\displaystyle\bigcup_{i=1}^\infty M_i$ is dense in $X.$ Then $\{M_i\}$ is a Schauder decomposition of $X$ if and only if there exists a constant $K \geq 1$ such that $\|\displaystyle \sum_{i \leq n} x_i\| \leq K \|\displaystyle \sum_{i \leq m} x_i \|$ for all $n, m$ with $n \leq m$ and for all sequences $\{x_i\}$ with $x_i \in M_i.$
Also we have a corollary of this theorem
Corollary
Let $\{v_i\}$ be a sequence of mutually orthogonal continuous projections of a Banach space $X$ (i.e. $v_i v_j = \delta_{ij} v_j$) such that $\overline{sp} \displaystyle \bigcup_{i=1}^\infty v_i(X) = X.$ Then $\{v_i(X)\}$ is a Schauder decomposition of $X$ if and only if there exists a constant $K \geq 1$ such that $\displaystyle \sup_n \|\displaystyle \sum_{i \leq n} v_i\| \leq K.$
So it is easily seen.
