An isometrically universal Banach space with a monotone Schauder basis
{\sc Joanna Garbuli\'nska-W\c{e
We present an isometric version of the complementably universal Banach space $\mathcal{B$ with a monotone Schauder basis. The space $\mathcal{B$ is isomorphic to Pe\l czy\'nski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property. \noindent MSC (2010) Primary: 46B04. % Isometric theory of Banach spaces Secondary: 46M15, % Categories, functors 46M40. % Inductive and projective limits \noindent Keywords: Monotone Schauder basis, linear isometry.
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\title{An isometrically universal Banach space with a monotone Schauder basis}
\author{
{\sc Joanna Garbuli\'nska-W\c{e}grzyn}\\ \\
{\small Institute of Mathematics,}
{\small Jan Kochanowski University (POLAND)}\\
{\small Faculty of Mathematics and Computer Science,} 
{\small Jagiellonian University (POLAND)}\\
{\small\texttt{jgarbulinska@ujk.edu.pl}}
\date{}
}
\begin{document}
\maketitle
\begin{abstract}
We present an isometric version of the complementably universal Banach space $\mathcal{B}$ with a monotone Schauder basis. The space $\mathcal{B}$ is isomorphic to Pe\l czy\'nski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property.

\noindent
\textbf{MSC (2010)}
Primary:
46B04. % Isometric theory of Banach spaces
Secondary:
46M15, % Categories, functors
46M40. % Inductive and projective limits

\noindent
\textbf{Keywords:} Monotone Schauder basis, linear isometry.
\end{abstract}

\section{Introduction}

\noindent A Banach space $X$ is \emph{complementably universal} for a given class of spaces if every space from the class is isomorphic to a complemented subspace of $X$.
In 1969 Pe\l czy\'nski \cite{pelbases} constructed a complementably universal Banach space with a Schauder basis.
In 1971 Kadec \cite{kadec} constructed a complementably universal Banach space for the class of spaces with the \emph{bounded approximation property} (BAP).
In the same year Pe\l czy\'nski \cite{pel_any} showed that every Banach space with BAP is complemented in a space with a basis.
Pe\l czy\'nski \& Wojtaszczyk  \cite{wojtaszczyk} constructed in 1971 a universal Banach space for the class of spaces with a finite-dimensional decomposition.
Applying Pe\l czy\'nski's decomposition argument \cite{pelczynski}, one immediately concludes that all three spaces are isomorphic.
It is worth mentioning a negative result of Johnson \& Szankowski \cite{JohnSzan} saying that no separable Banach space can be complementably universal for the class of all separable spaces.
The author in \cite{asia} presented a natural extension property that describes an isometric version of the Kadec-Pe\l czy\'nski-Wojtaszczyk space. The constructed space is unique, up to isometry, for the class of Banach spaces with finite-dimensional decomposition and isomorphic to the Kadec-Pe\l czy\'nski-Wojtaszczyk space.

In this note we present an isometric version of the complementably universal Banach space with a monotone Schauder basis. Most of the arguments are inspired by the recent works \cite{asia}, \cite{kubis} and \cite{solecki}.

\section{Preliminaries}
A \emph{projectional resolution of the identity} (briefly: \emph{PRI}) on a Banach space $X$ is a sequence of norm-one projections $\{P_n\}_{n\in \omega}$ of $X$ satysfying following conditions:
\begin{enumerate}
	\item[(1)] $P_n \circ P_m = P_{min\{n,m\}} = P_m \circ P_n$ for every $n, m \in \omega$;
	\item[(2)] $\operatorname{dim}(P_n[X])=n$;
	\item[(3)] $X=\operatorname{cl}\bigcup _{n\in \omega} P_n[X]$.
\end{enumerate}

A \emph{Schauder basis} is a sequence $\{e_n\}_{n\in \omega}$ of vectors in a Banach space $X$ such that for every $x\in X$ there are uniquely determined scalars $\{a_n\}_{n\in \omega}$ such that
$$x=\sum_{n=0}^{\infty}a_n e_n,$$
where the convergence of the series is taken with respect to the norm. Once this happen, for each $n\in \omega$ there is a cannonical projection $P_n$ defined by
$$P_n(\sum_{i\in \omega}a_i e_i)=\sum_{i<n} a_i e_i.$$ 

A Banach space $X$ has a \emph{monotone Schauder basis} if and only if it has a PRI $\{P_n\}_{n\in \omega}$.

On the other hand, the basis is \emph{monotone} if $\|P_n \|\leq 1$ for every $n\in \omega$.  

Recall that a Banach space $X$ is \emph{1-complemented} in $Y$ if there exists a projection $P:Y\to X$ such that $\|P\| \leq 1$ and $P[Y]=X$. 

Given Banach spaces $Y\subseteq X$, we say that $Y$ is an \emph{initial subspace} of $X$ if there is a sequence of norm-one projections $\{P_n\}_{n\in \omega}$ satysfying conditions (1), (3) and
\begin{enumerate}
    \item[$1^{\circ }$] for each $n\in \omega$ the image $P_{n+1}-P_n$ is 1-dimensional,
	\item[$2^{\circ }$] $X=P_0[Y]$.
\end{enumerate} 
Typical examples of initial subspaces are linear spans of initial parts of a Schauder basis.
Note that, an initial subspace is 1-complemented and the trivial space is initial in $Y$ if and only if $Y$ has a monotone Schauder basis.

Given a Schauder basis $\{e_n\}_{n\in \omega}$ in $X$, given a subset $S\subset \omega$, we say that $\{e_n\}_{n\in S}$ is a \emph{cannonically 1-complemented subbasis} if the linear operator $P_S:X\to X$ defined by conditions $P_Se_n=e_n$ for $n\in S$, $P_Se_n=0$ for $n\notin S$, has norm $\leq $ 1.

Finally, we say that a basis $\{v_n\}_{n\in \omega}$ is \emph{isometric} to a subbasis of $\{e_m\}_{m\in \omega}$ if there is an increasing function $\varphi :S\to \omega$ such that the linear operator $f$ defined by equations $f(v_n)=e_{\varphi (n)}$ ($S\subseteq \omega$) is a linear isometric embedding. 

Every finite-dimensional Banach space $E$ is isometric to $\R^n$ with some norm $\| \cdot \|$.
We shall say that $E$ is \emph{rational} if $E = \R^n$ with a norm such that its unit ball is a polyhedron spanned by finitely many vectors whose every coordinate is a rational number.
Equivalently, $X$ is rational if, up to isometry, $X = \R^n$ with a ``maximum norm" $\| \cdot \|$ induced by finitely many functionals $\varphi_0,\dots, \varphi_{m-1}$ such that $ \varphi_i[\Q^n] \subseteq  \Q$ for every $i < m$. More precisely, $$\| x\| = \max_{i < m}{\mid \varphi_i(x)\mid }$$
for $x\in \R^n$.
Typical examples of rational Banach spaces are $\ell_1(n)$ and $\ell_{\infty}(n)$, the $n$-dimensional variants of $\ell_1$ and $\ell_{\infty}$, respectively.
On the other hand, for $1 < p < \infty$, $n > 1$, the spaces $\ell_p(n)$ are not rational.
Of course, every rational Banach space is polyhedral.
An operator $T: {\R^n\to \R^m}$ is \emph{rational} if $T[\Q^n] \subseteq  \Q^m$.

It is clear that there are (up to isometry) only countably many rational Banach spaces and for every $\eps > 0$, every finite-dimensional space has an $\eps$-isometry onto some rational Banach space.

Let $X$ is a set. A {\it convex hull} is the minimal convex set containing $X$.

Let $\K$ be a category and $A,B\in \K$. By $\K(A,B)$ we will denote the set of all $\K$-morphisms from $A$ to $B$. A \emph{subcategory} of $\K$ is a category $\DL$ such that each object of $\DL$ is an object of $\K$ and each arrow of $\DL$ is an arrow of $\K$. 

Category $\DL$ is \emph{cofinal} in $\K$ if for every $A\in \K$ there exist an object $B\in \DL$ such that the set $\K(A,B)$ is nonempty. Let $\K$ be a category. $\K$ has the {\it amalgamation property} if for every objects $A, B, C\in \K$ and for every morphisms $f\in \K(A,B)$, $g\in \K(A,C)$ we can find object $D\in \K$ and morphisms $f'\in \K(B,D)$, $g'\in \K(C,D)$ such that $f'\circ f=g'\circ g$.

Category $\K$ has the \emph{joint embedding property} if for every objects $A, B\in \K$ we can find some object $C\in \K$ such that there exist morphisms $f\in \K(A,C)$, $g\in \K(B,C)$.

\section{The Amalgamation}

\begin{lemma}\label{amalgamacja} {\bf (Amalgamation Lemma)} Let $Z$, $X$, $Y$ be finite-dimensional Banach spaces, such that $i:Z\to X$, $j:Z\to Y$ are isometric embeddings and $\{0\}$, $i[Z]$, $j[Z]$ are initial subspaces of $Z$, $X$, $Y$, respectively. Then there exists a finite-dimensional Banach space $W$, isometric embeddings $i':X\to W$, $j':Y\to W$ such that $i'[X]$, $j'[Y]$ are initaial subspaces of $W$ and the following commutes:  
$$\xymatrix{
Y  \ar[r]^{j'} & W \\
 Z  \ar[r]^i \ar[u]^j & X.\ar[u]^{i'}}$$
\end{lemma}

\begin{proof} 
Let $Z$, $X$, $Y$ be finite-dimensional Banach spaces ($\operatorname{dim}(Z)=N$, $\operatorname{dim}(X)=M$,  $\operatorname{dim}(Y)=K$ and $N\leq M, K$), where $\{0\}$ is an initial subspace $Z$, $i[Z]$ is an initial subspace of $X$ and $j[Z]$ is an initial subspace of $Y$, with sequences of norm-one projections $\{P_n\}_{n\leq N}$, $\{Q_n\}_{n\leq M-N}$, $\{R_n\}_{n\leq K-N}$, respectively. Observe that $Q_0[X]:=i(Z)=Z$ and $R_0[Y]:=j(Z)=Z$ (we assume that $i, j$ are inclusions).

We define $W$ as a $(X\oplus Y)/\Delta $, where $\Delta =\{(z,-z):z\in Z\}$.
Given $(x,y)\in X\oplus Y$, define a norm $\| (x,y)\| =\| x\| _X+\| y\| _Y$.
Let $i'_X(x)=(x,0)+\Delta $ and $j'_Y(y)=(0,y)+\Delta .$
Then $i', j'$ are isometric embeddings (see \cite{asia} or \cite{kubis}).

We have to show that $i'[X], j'[Y]$ are initial subspaces of $W$ (observe that $\operatorname{dim}(W)=M+K-N$). We will definie sequences of projections $\{S_n\}_{n\leq M}$, $\{T_n\}_{n\leq K}$ such that $S_0:=i'\circ Q_{M-N}=(Q_{M-N},0)+\Delta$ and $T_0:=j'\circ R_{K-N}=(0,R_{K-N})+\Delta$. Note that $Q_0[X]=R_0[Y]$.
Define $S_n(x,y):=(Q_{M-N}(x),R_{n}(y))+\Delta $ for $n\geq 1$.
Similarly, $T_n(x,y):=(Q_{n}(x),R_{K-N}(y))+\Delta $ for $n\geq 1$.

It is easy to show that $\| S_n\| \leq 1$ and $\| T_n \| \leq 1$.
Observe that:
\begin{enumerate}
\item
$\operatorname{dist}((Q_{M-N}(x),R_{n}(y))+\Delta)\leq \|(Q_{M-N}(x),R_{n}(y))\| =\|Q_{M-N}(x)\|_X+\|R_{n}(y)\|_Y \leq \|x\|_X + \|y\|_Y$;
\item
$\operatorname{dist}((Q_{n}(x),R_{K-N}(y))+\Delta )\leq \|(Q_{n}(x),R_{K-N}(y))\|= \|Q_{n}(x)\|_X+\|R_{K-N}(y)\|_Y \leq \|x\|_X + \|y\|_Y$.
\end{enumerate}
This completes the proof
\end{proof}


Fix $\eps > 0$ and fix a linear operator $f:X\to Y$ such that
$$(1 + \eps)^{-1}\cdot  \| x\| \leq  \|f(x)\| \leq  \| x\|$$
for $x \in X$ and $f[X]$ is an initial subspace of $Y$.
Consider the following category $\K_f^{\eps}$.
The objects of $\K_f^{\eps}$ are pairs $(i, j)$ of linear operators $i: X\to Z$, $j:Y\to Z$ between Banach spaces with initial subspaces such that
\begin{enumerate}
	\item $i[X]$, $j[Y]$ are initial subspaces of $Z$; 
	\item $\|i\| \leq  1$ and $\| j \| \leq  1$;
	\item $\|i(x) - j(f(x))\| \leq  \eps \cdot \| x\|$ for $x \in X$.
\end{enumerate}

Given $a_0 = (i_0,j_0)$ and $b_0 = (i_1,j_1)$ in $\K_f^{\eps}$, where $i_k: X\to {Z_k}$ and $j_k: Y\to {Z_k}$ for $k < 2$, an arrow from $a_0$ to $b_0$ is defined to be a linear operator $T:{Z_0}\to {Z_1}$ such that $\|T\| \leq  1$, $T \circ  i_0 = i_1$, and $T \circ  j_0 = j_1$.

\begin{lemma}\label{poprawianie}
The category $\K_f^{\eps}$ has an initial object $(i_0,j_0)$ such that both $i_0$, $j_0$ are canonical isometric embeddings into $X \oplus Y$ with a suitable norm $\|\cdot \|$ and $X \oplus Y$ has initial subspaces $X$, $Y$.
\end{lemma}

\begin{proof}
Define
$$G = \{(x, -f(x)) \in X \times Y: x \in  \eps^{-1} B_X\}.$$
Recall that $B_X$ and $B_Y$ are the unit balls of $X$ and $Y$ respectively.
Finally, let $K$ be the convex hull of $G \cup (B_X \times \{0\}) \cup (\{0\} \times B_Y)$ and let  $$\| (x,y) \| _K=\inf\{\| u\| _X+\|v \| _Y+\eps \|w \| _X :(x,y)=(u,0)+(0,v)+(w,-f(w)), (x,y)\in K\}$$ on $K$.
We claim that $\| \cdot \|_K$ is as required.\\
Define linear operators $i_0(x)=(x,0)$ and $j_0(y)=(0,y)$.  We check that $\| i_0(x)-j_0(f(x))\|_K\leq ~\eps \|x\|_X$.

We have that $\|(x,-f(x))\|_K \leq \eps \|x\|_X$. This implies that 
$$\|i_0(x)-j_0(f(x))\|_K = \|(x,-f(x))\|_K \leq \eps \|x\|_X.$$
This proves that $(i_0,j_0)$ is an object of the category $\K_f^{\eps}$.

Let $ (w,-f(w))\in G$, $ (u,0)\in  B_X \times \{0\}$, $(0,v)\in \{0\} \times  B_Y$ and let $\| u\| \leq 1$, $\|v\| \leq 1$, $\|w\| \leq \eps^{-1}$. Then $ (x,y)\in K$ is a linear combination:
\begin{align*}
(x,y)&=(u,0) + (0,v) + (w,-f(w))=(u + w,v - f(w)).
\end{align*}

Now we prove that $\|(x,0)\|_K=\|x\|_X$ and $\|(0,y)\|_K=\|y\|_Y$. 
Suppose that $\|x\|=1$ and $y=0$; then $v=f(w)$.

It is easy to show that $\|(x,0)\|_K\leq \|x\|_X$ (we take $u=x$, then $w=0$ and $v=f(w)=0$). Observe that
\begin{align*} 
&\|u\|_X +\|v\|_Y+\eps \|w\|_X= \|u\|_X +\|f(w)\|_Y+\eps \|w\|_X\geq \\ &\geq \|u\|_X +(1-\eps)\|w\|_X+\eps \|w\|_X = \|u\|_X +\|w\|_X \geq \|x\|_X .
\end{align*}

Suppose that $x=0$ and $\|y\|=1$, then $u=-w$. It is obvious that $\|(0,y)\|_K\leq \|y\|_Y$ (we take $v=y$, then $f(w)=0$ and $u=-w=0$). On the other hand
\begin{align*}
& \|u\|_X +\|v\|_Y+\eps \|w\|_X= \|w\|_X +\|v\|_Y+\eps \|w\|_X= \\ &= (1+\eps)\|w\|_X +\|v\|_Y\geq (1+\eps) \|w\|_X+\|v\|_Y \geq  \|f(w)\|_Y +\|v\|_Y\geq \|y\|_Y.
\end{align*}
This proves that $i_0$, $j_0$ are isometric embeddings.\\

We have to check that the convex hull $K$ is a unit ball of the norm $\| \cdot \|_K$.
Let $$B_K=\{(x,y): \|(x,y)\|_K\leq 1\}.$$
The inclusion $K\subseteq B_K$ is obvious. To prove that $B_K\subseteq K$, fix $(x,y)$ such that $\|(x,y)\|_K<~1$. Then 
$$\|u\|_X+\|v\|_Y+\eps \|w\|_X<1$$ 
for some $u,v,w$ such that $x=u+w$ and $y=v-f(w)$.
Observe that $(u,0)\in B_X\times \{0\}$,  $(0,v)\in \{0\}\times B_Y$, $(w,-f(w))\in G$ and $(u,0)+(0,v)+(w,-f(w))=(x,y)$.

Denote $\alpha =\|u\|_X+\|v\|_Y+\eps \|w\|_X$, then $\alpha <1$.
Let $\lambda _1= \frac{\|u\|_X}{\alpha }$, $\lambda _2= \frac{\|v\|_Y}{\alpha}$ and $\lambda _3= \frac{\eps \|w\|_X}{\alpha}$.
Then $(u,0)=\lambda_1 (\frac{\alpha}{\|u\|_X}(u,0))\in B_X \times \{0\}$, $(0,v)=\lambda_2(\frac{\alpha}{\|v\|_Y}(0,v))\in \{0\}\times B_Y$ and  $(w,-f(w))=\lambda_3(\frac{\alpha}{\eps \|w\|_X}(w,-f(w)))\in G$.

We have to check that the pair $(i_0,j_0)$ is an initial object.
This means that for every $(i,j) \in Obj(\K_f^{\eps})$ there exists
a unique linear operator $T:X\oplus Y\to Z$ such that $T\circ
i_0=i$, $T\circ j_0=j$ and the norm of $T$ is less or equal to 1.

Fix $(i,j) \in Obj(\K_f^{\eps})$ and define $T(x,y)=i(x)+j(y)$. 
It is clear that this is the only possibility for $T$.
We will check that $\|T\|_Z\leq 1$.

Let $(x,y)\in K$, then 
\begin{enumerate}
\item[$1^{\circ}$]  if $ (x,y)\in B_X \times\{0\}$ then $\|T(x,0)\|_Z=\| i(x)\|_Z \leq  1$,
\item[$2^{\circ}$] if $ (x,y)\in \{0\} \times B_Y$ then $\| T(0,y)\|_Z=\|j(y)\|_Z \leq  1$,
\item[$3^{\circ}$]  if $ (x,y)\in G$ then $\|T(x,-f(x))\|_Z=\|i(x)-j(f(x))\|_Z\leq \eps \cdot \|x\|_Z\leq \eps \cdot \eps^{-1}=1$.
\end{enumerate}
 
Let $X$ and $Y$ be finite-dimensional Banach spaces ($\operatorname{dim}(X)=M$,  $\operatorname{dim}(Y)=K$ and $M\leq K$), where $\{0\}$ is an initial subspace $X$ and $f[X]$ is an initial subspace of $Y$, with sequences of norm-one projections $\{P_n\}_{n\leq M}$ and $\{Q_n\}_{n\leq K-M}$, respectively. Observe that $Q_0[Y]:=f(X)$.

We prove that $X, Y$ are initial subspaces of $X \oplus Y$ ($\operatorname{dim}(X\oplus Y)=M+K$). We define sequences of projecions $\{R_n\}_{n\leq K}$, $\{S_n\}_{n\leq M}$ such that $R_0:=i\circ P_M$ and $S_0:=j\circ Q_{K-M}$.

%Mark $P_{M-1}:=\bigcup_{n\leq M-1} P_n$ and $Q_{K-M-1}:=\bigcup_{n\leq K-M-1} Q_n$.

Let $R_{n}:= i\circ P_{M}+j\circ f\circ P_{n} = (P_{M},f\circ P_{n})$ for $n\leq M$ %(without $f(P_{0})$, which is 0-dimensional) 
and $R_{M+1+n}:= i\circ P_{M}+j\circ Q_{n+1}= (P_{M},Q_{n+1})$ for ${n\leq K-M-1}$.
Similarly $S_{n}:= i\circ P_{n}+j\circ Q_{K-M} = (P_{n}, Q_{K-M})$ for $n\leq M$.


It is easy to show that $\|R_n\|_K\leq 1$ and $\|S_n \|_K \leq 1$.
Observe that if
\begin{enumerate}
\item[$1^{\circ}$]
 $\|(P_{M},f\circ P_{n})\| \leq 1$ for $n\leq M$, we take $u=P_{M}$, $v=0$, $w=0$, $f(w)=f\circ P_{n}$.
 \item[$2^{\circ}$]
 $\|(P_{M},Q_{n+1})\| \leq 1$ for ${n\leq K-M-1}$, we take $u=P_{M}$, $v=0$, $w=0$, $f(w)=Q_{n+1}$.
\item[$3^{\circ}$]
$\|(P_{n}, Q_{K-M})\| \leq 1$ for $n\leq M$, we take $u=P_{n}$, $v=0$, $w=0$, $f(w)=Q_{K-M}$.

\end{enumerate}
\end{proof} 
First version of the proof of the lemma about extending $\eps$-isometry between Banach spaces can be found in \cite{asia} and \cite{solecki}.



\section{A construction}
In order to make some statements shorter, we shall consider $1$-bounded operators, this means linear operators of norm at most 1 only.

We shall now prepare the setup for our construction.

We now define the relevant category $\K$.
The objects of $\K$ are finite-dimensional Banach spaces.
Given finite-dimensional spaces $X$, $Y$, an $\K$-arrow is an isometric embedding $f:X\to Y$ such that $f[X]$ is an initial object of $Y$ and $\{0\}$ is an initial object of $X$ (both have a monotone Schauder basis).

Denote by $\DL$ the subcategory of $\K$ consisting of all rational $\K$-arrows.
Obviously,  $\DL$ is countable.
Looking at the proof of Lemma~\ref{amalgamacja}, we can see that $\DL$ has the amalgamation property.
We now use the concepts from \cite{kubis} for constructing a ``generic" sequence in $\DL$.
First of all, a \textit{sequence} in a fixed category $\DC$ is formally a covariant functor from the set of natural numbers $\omega$ into $\DC$.


Up to isomorphism, every sequence in $\DL$ corresponds to a chain $\{X_n\}_{n\in \omega}$ of finite-dimensional subspaces with initial subspaces.
By this way, the monotone Schauder basis of a Banach space $X$ is translated into the existence of a sequence in $\K$ whose co-limit is $X$.
For the sake of convenience, we shall denote a sequence by $\vec  U$, having in mind a chain $\{ U_n\}_{n \in \omega}$ of finite-dimensional spaces with the initial subspaces.
Given $U_n \subseteq  U_m$, the $\K$-arrow $f^m_n:U_n\to U_m$ is such that the image $f^m_n[U_n]$ is an initial subspace of $U_m$.

Following \cite{kubis}, we shall say that a sequence $\vec Y$ in $\DL$ is \textit{Fra\"iss\'e}
if it satisfies the following condition:
\begin{enumerate}
\item[(A)] Given $n\in \omega$, and an $\DL$-arrow $f: {Y_n}\to Z$, there exist $m > n$ and an $\DL$-arrow $g: Z\to {Y_m}$ such that $g \circ  f$ is the arrow from $Y_n$ to $Y_m$.
\end{enumerate}
It is clear that this definition is purely category-theoretic.
The name ``Fra\"iss\'e sequence", as in \cite{kubis}, is motivated by the model-theoretic theory of Fra\"iss\'e limits explored by Roland Fra\"iss\'e~\cite{fraise}.
One of the results in \cite{kubis} is that every countable category with amalgamations has a Fra\"iss\'e sequence.

\begin{theorem}[\cite{kubis}]\label{ciag}
The category $\DL$ has a Fra\"iss\'e sequence.
\end{theorem}

From now on, we fix a Fra\"iss\'e sequence $\{Y_n\}_{n\in \omega}$ in $\DL$.
As usual, we assume that the embeddings are inclusions.
Let $\B$ be the completion of the union $\bigcup_{n\in \omega}Y_n$.

\section{Universality}



\begin{theorem}\label{ThmUnivP}
Let $X$ be a Banach space with a monotone Schauder basis.
Then there exists an isometric embedding $e:X\to \B$ such that $e[X]$ is an initial subspace of $\B$.
\end{theorem}

\begin{proof}
%See the proof in \cite[Theorem 5.1]{asia}.
Fix a Banach space $X$ with a monotone Schauder basis and let this be witnessed by a chain $\{ X_n\}_{n\in \omega}$ together with suitable projections $\{Q_n\}_{n\in \omega}$.

We construct inductively $1$-bounded operator $e_n:X_n\to Y_{k_n}$ such that
\begin{enumerate}
	\item[(1)] $e_n[X_n]$ is an initial subspace of $Y_{k_n}$,
	\item[(2)] $\| e_{n+1} \restriction X_n - e_n\| < 2^{-n}$.
\end{enumerate}
Recall that, according to our previous agreement, we consider only $1$-bounded operators.
We may assume that $X_0 = Y_0 = \{0\}$, therefore it is clear how to start the induction.
Suppose $e_n$ (and $k_n \in \omega$) have already been defined.
By Lemma~\ref{poprawianie}, there exist $i:X_n\to ~W$ and $j: Y_{k_n}\to ~W$, where $W = X_n \oplus_{e_n} Y_{k_n}$, and the following conditions are satisfied:
\begin{enumerate}
	\item[(3)] $\|j \circ e_n - i\| < 2^{-n}$.
\end{enumerate}
Using Lemma~\ref{amalgamacja}, we may further extend $W$ so that there exists also a  $\ell: X_{n+1}\to W$ satisfying
\begin{enumerate}
	\item[(4)] $\ell \restriction X_n = i$.
\end{enumerate}
Applying Lemma~\ref{poprawianie} and Lemma~\ref{amalgamacja} we preserve condition (1).
Recall that $Y_n$ is a rational Banach space. Thus, we can extend $W$ further, so that the extended arrow from $Y_n$ to $W$ will become rational.
Doing this, we make some ``error" of course, although we can still preserve (3) and (4), because all the inequalities appearing there are strict.
Now we use the fact that $\{ Y_n \}_{n\in \omega}$ is a Fra\"iss\'e sequence.
Specifically, we find $k_{n+1} > k_n$, rational operators $g:W\to {Y_k}$ and $H: Y_{k_{n+1}}\to W$ such that $H \circ g = \operatorname{id}_W$ and $g \circ j$ is the inclusion $Y_{k_n} \subseteq Y_k$.

Define $e_{n+1} = g \circ \ell$.
This finishes the inductive construction.

Passing to the limits, we obtain $1$-bounded operator $e:X\to \B$ .
Condition (2) imply that $e$ is an isometric embedding.
In particular, $e[X]$ is an initial subspace of $\B$.

\end{proof}

\begin{corollary}
The space $\B$ is isomorphic to Pe\l czy\'nki's complementably universal space for Schauder bases, as well as to Kadec's complementably universal space for the bounded approximation property.
\end{corollary}

\begin{proof}
See the proof in \cite[Corollary 5.2]{asia}.
\end{proof}



\section{Isometric uniqueness}

Proofs are done analogous like in \cite{asia}. We present only the theorems and references, when we can find the proofs.

Let us consider the following extension property of a Banach space $E$:
\begin{enumerate}
	\item[(B)] Given a pair $X\subseteq  Y$ of finite-dimensional Banach spaces with monotone Schauder basis such that $\{0\}$, $X$ are initial subspaces of those spaces, respectively, given an isometric embedding $f:X\to E$ such that $f[X]$ is an initial subspace of $E$, for every $\eps > 0$ there exists an $\eps$-isometric embedding $g:Y\to E$ such that $\| g \restriction X - f\| < \eps$ and $g[Y]$ is an initial subspace of $E$.
\end{enumerate}

\begin{theorem}
$\B$ satisfies condition (B).
\end{theorem}

\begin{proof}
See the proof in \cite[Theorem 6.1]{asia}.
%We use the fact that every finite dimensional Banach space can be approximated by a rational Banach space. Next we use Lemma~\ref{poprawianie} and Lemma~\ref{amalgamacja}.
\end{proof}

\begin{lemma}
Assume $X$ satisfies condition (B).
Then, given $\eps, \delta > 0$, given finite-dimensional spaces $E \subseteq  F$, given an $\eps$-isometric embedding $f:E\to X$ such that $f[E]$ is an initial subspace of $X$, there exists a $\delta$-isometric embedding $g:F\to X$ such that $\| g\restriction E - f\| < \eps$ and $g[F]$ is an initial subspace of $X$.
\end{lemma}

\begin{proof}
See the proof in \cite[Lemma 6.2]{asia}.
\end{proof}

\begin{theorem}
Let $\B$ and $\bK$ be Banach spaces with monotone Schauder bases satisfying condition (B) and let $h:A\to B$ be a bijective linear isometry between finite-dimensonal subspaces, where $A$ and $B$ are initial subspaces of $\B$ and $\bK$, respectively.
Then for every $\eps > 0$ there exists a bijective linear isometry $H: \B \to \bK$ that is $\eps$-close to $h$.
In particular, $\B$ and $\bK$ are linearly isometric.
\end{theorem}

\begin{proof}
See the proof in \cite[Theorem 6.3]{asia}. We use standard back-and-forth argument.
Instead of codition (4) from proof of \cite[Theorem 6.3]{asia}, we have that $f_n[A_n]$ and $g_n[B_n]$ are initial subspaces of $\bK$ and $\B$, respectively.
\end{proof}

\begin{thebibliography}{}
\bibitem{fraise} \textsc{R. Fra\"iss\'e}, \textit{Sur quelques classifications des syst\`emes de relations}, Publ. Sci. Univ. Alger. S\'er. A. \textbf{1} (1954) 35--182

\bibitem{asia} \textsc{J. Garbuli\'nska}, \textit{Isometric uniqueness of a complementably universal Banach space for Schauder decompositions}, Banach J. Math. Anal. 8 (2014), no. 1, 211--220

\bibitem{JohnSzan} {\sc W.B. Johnson, A. Szankowski}, \textit{ Complementably universal Banach spaces}, Studia Math. \textbf{58} (1976) 91--97

\bibitem{kadec} {\sc M. I. Kadec}, \textit{ On complementably universal Banach spaces}, Studia Math. \textbf{40} (1971) 85--89

\bibitem{k77} \textsc{N. Kalton}, \textit{Universal spaces and universal bases in metric linear spaces}, Studia Math. 61 (1977), 161--191

\bibitem{kubis} \textsc{W. Kubi\'s}, \textit{Fra\"{\i}ss\'{e} sequences: category-theoretic approch to universal homogeneus structures}, preprint, \href{http://arxiv.org/abs/0711.1683}{arxiv.org/abs/0711.1683}

\bibitem{solecki} \textsc{W. Kubi\'s, S. Solecki}, \textit{A proof of uniqueness of the Gurari\u\i\/ space}, Israel J. Math. 195 (2013), 449--456

\bibitem{pel_any} \textsc{A. Pe{\l}czy\'nski}, \textit{Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis},
Studia Math. \textbf{40} (1971) 239--243

\bibitem{pelczynski} \textsc{A. Pe\l{}czy\'nski}, \textit{Projections in certain Banach spaces}, Studia Math. 19 (1960), 209-228

\bibitem{pelbases} \textsc{A. Pe\l{}czy\'nski}, \textit{Universal bases}, Studia Math. 32 (1969), 247-268

\bibitem{wojtaszczyk} \textsc{A. Pe\l{}czy\'nski, P. Wojtaszczyk}, \textit{Banach spaces with finite-dimensional expansions of identity and universal bases of finite-dimensional subspaces}, Studia Math. \textbf{40} (1971) 91--108
\end{thebibliography}

\end{document}

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