$x\in H_k\left(\{e_n\}\right)$ can be uniquely represented in a series $$x=\sum\limits_{n=1}^{\infty} x_n,$$ where $x_n\in \mathfrak{M}_n$ for each $n\in\mathbb{N}$, and there exists an associated sequence of coordinate linear projections $\{P_n\}_{n=1}^{\infty}$ defined by $P_n x=P_n \sum\limits_{m=1}^{\infty} x_m=x_n,$ where $x_n\in \mathfrak{M}_n,$ $n\in\mathbb{N}$. It follows that for every $n,j\in\mathbb{N}$ $$\label{proj}
    P_n e_j =\left \{
\begin{array}{l}
e_j,\quad j\in A_n,\\
0,\quad \:j\notin A_n.\\
\end{array}
\right .$$

Consider an element $x^{\ast}=(\mathfrak{f})\sum\limits_{j=1}^{\infty}e_j\in H_k\left(\{e_n\}\right).$ Then, taking into account (\[proj\]), we have that for every $n\in\mathbb{N}$ $$\label{pn_proj}
   P_n x^{\ast}= P_n \left((\mathfrak{f})\sum\limits_{j=1}^{\infty}e_j\right)= \sum\limits_{j\in A_n} e_j.$$

We recall that the norm in a Hilbert space $H_k\left(\{e_n\}\right)$ is defined by $$\begin{aligned}
\|x\|_k &=\left\|(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\right\|_k\\
&=\left\|\sum\limits_{n=1}^{\infty}\left(c_{n} -C_{k}^{1} c_{n-1}+\dots+  (-1)^{k+1} C_{k}^{k-1} c_{n-k+1} +(-1)^{k} c_{n-k}\right) e_n\right\|,\end{aligned}$$ where $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\in H_k\left(\{e_n\}\right),$ $C_k^m$ are binomial coefficients, $\|\cdot \|$ denotes the norm in an initial Hilbert space $H$ and $c_{1-j}=0$ for all $j\in\mathbb{N}$, see [@Sklyar3]. Since $\{e_n\}_{n=1}^{\infty}$ is a Riesz basis of $H$ (see Theorem \[construction\]), there exist two constants $M\geq m>0$ such that for every $y=\sum\limits_{n=1}^{\infty} \alpha_n e_n\in H$ we have $$\label{1Riesz}
  m \|y\|^2 \leq \sum\limits_{n=1}^{\infty} |\alpha_n|^2 \leq M \|y\|^2.$$

By virtue of (\[pn\_proj\]) and (\[1Riesz\]) we obtain that for every $n\in\mathbb{N}$ $$\left\|P_n x^{\ast} \right\|_k^2=\left\|\sum\limits_{j\in A_n} e_j \right\|_k^2=\left\|(\mathfrak{f})\sum\limits_{j=1}^{\infty}\xi_j(n) e_j \right\|_k^2\geq \frac{1}{M},$$ where for every $n,j\in\mathbb{N}$ $$\xi_j(n)=\left \{
\begin{array}{l}
1,\quad 