\begin{document}

v^\astitle[On the Banach-Mazur distance between continuous function spaces]
{On the Banach-Mazur distance between continuous function spaces with scattered boundaries}

\author{Jakub Rondo\v s}

\address{Charles University\\
Faculty of Mathematics and Physics\\
Department of Mathematical Analysis \\
Sokolovsk\'{a} 83, 186 \ 75\\mathcal Praha 8, Czech Republic}

e^\astmail{jakub.rondos@gmail.com}

u^\astubjclass[2010]{46B03; 46A55; 46E40}

\keywords{function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary}

v^\asthanks{The research was supported by the Research grant GAUK 864120}

\begin{abstract}
We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon \cite{Gordon3}, we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal C(K, E)$ spaces.
e^\astnd{abstract}

\maketitle

u^\astection{Introduction}

For a locally compact (Hausdorff) space $K$ and a real Banach space $E$, let $\mathcal C_0(K,E)$ denote the space of all continuous $E$-valued functions vanishing at infinity endowed with the sup-norm
\[\norm{f}_{u^\astup}=u^\astup_{x \in K} \norm{f(x)}, \quad f \in \mathcal C_0(K,E).\]
If $K$ is compact, then this space will be denoted by $\mathcal C(K,E)$. We write $\mathcal C_0(K)$ for $\mathcal C_0(K, e^\astr)$ and $\mathcal C(K)$ for $\mathcal C(K, e^\astr)$. All unexplained notions and definitions are contained in the next section.

We start with the following generalization of the well-known Banach-Stone theorem given independently by Amir \cite{amir} and Cambern \cite{cambern}. They showed that compact spaces $K_1$ and $K_2$ are homeomorphic if there exists an isomorphism $T\operatorname{co}lon \mathcal C(K_1)v^\asto\mathcal C(K_2)$ with $\norm{T} \norm{T^{-1}}<2$. Moreover, Amir conjectured that the number $2$ may be replaced by $3$. Cohen \cite{cohen-bound2} showed that this conjecture is not true in general. However, in \cite{Gordon3}, Gordon proved that it is true in the class of countable compact spaces: If $K_1$, $K_2$ are nonhomeomorphic countable compact spaces and $T: \mathcal C(K_1) |ightarrow \mathcal C(K_2)$ is an isomorphism, then $\norm{T}\norm{T^{-1}} \geq 3$.

The result of Gordon was extended in \cite[Theorem 1.5]{CandidoGalego3}, where the authors show that if $E$ is a Banach space having non-trivial cotype, and such that for every $n \in e^\astn$, $E^n$ contains no subspace isomorphic to $E^{n+1}$, then countable compact spaces $K_1$ and $K_2$ are homeomorphic provided there exists an isomorphism $T:\mathcal C(K_1, E) |ightarrow \mathcal C(K_2, E)$ with $\norm{T}\norm{T^{-1}}<3$. It is clear that every finite-dimensional Banach space satisfies the above condition, and the authors also show in \cite[Remark 4.1]{CandidoGalego3} that there exist many infinite-dimensional Banach spaces that satisfy it. 

Starting in \cite{chuco}, and continuing in \cite{lusppams}, \cite{dosp}, and \cite{rondos-spurny-spaces}, the theorem of Amir and Cambern was extended to the context of subspaces. The final result for subspaces of scalar functions (see \cite[Theorem 1.1]{rondos-spurny-spaces}), reads as follows. For $i=1,2,$ let $\mathcal H_i u^\astubseteq \mathcal C_0(K_i)$ be closed subspaces such that all points in their Choquet boundaries are weak peak points. If there exists an isomorphism $T\operatorname{co}lon \mathcal H_1v^\asto \mathcal H_2$ with $\norm{T} \norm{T^{-1}}<2$, then their Choquet boundaries $\mathcal Ch_{\mathcal H_i} K_i$ are homeomorphic (we recall that $x\in K_i$ is a e^\astmph{weak peak point} (with respect to $\mathcal H_i$) if for a given $e^\astp\in (0,1)$ and a neighborhood $U$ of $x$ there exists a function $h\in B_{\mathcal H_i}$ such that $h(x)>1-e^\astp$ and $\abs{h}<e^\astp$ on $\mathcal Ch_{\mathcal H_i} K_iu^\astetminus U$). Vector-valued extension of this result was given in \cite{rondos-spurny-vector-spaces}.

For a closed subspace $\mathcal H$ of $\mathcal C(K, E)$ we denote by $\mathcal W_{\mathcal H}$ the set of weak peak points of $K$ with respect to $\mathcal H$ (for the definition, which is similar to the one for scalar subspaces, see next section), and we consider the set
\[\Omega_{\mathcal H}=\{ y \in K, e^\astxists g \in \mathcal H \operatorname{co}lon g(y) \neq 0\}.\] 
The notion of the set $\Omega_{\mathcal H}$ will be useful, as among other things, it will allow us to work efficiently at the same with functions defined on compact spaces as well as on locally compact spaces. We also note that since $\Omega_{\mathcal H}$ is an open set in $K$, if $F^{(\alpha)} \cap \Omega_{\mathcal H}$ is finite for a subset $F$ of $K$ and an ordinal $\alpha$, then 
$F^{(\alpha+1)} \cap \Omega_{\mathcal H}$ is empty. A set $\mathcal B u^\astubseteq K$ is a e^\astmph{boundary} for $\mathcal H$ if for each $h \in \mathcal H$, $\norm{h}=u^\astup_{x \in \mathcal B} \norm{h(x)}$.

In view of the above results for subspaces, it seems natural to be looking for an extension of the result of Gordon to this context. We obtain such an extension as a corollary of the following theorem, that is inspired by \cite[Theorem 2.1]{CandidoGalego3} and \cite[Theorem 1.7]{GalegoVillamizar}. Also note that a similar result has been proved in \cite[Theorem 4]{VillamizarFelipe} for the case when $\mathcal H_1$ is an extremely regular subalgebra of $\mathcal C_0(K_1)$ and $\mathcal H_2=\mathcal C_0(K_2, E)$ (we recall that a subspace $\mathcal H u^\astubseteq \mathcal C_0(K)$ is e^\astmph{extremely regular} if for each $x \in K$, $U$ open neighbourhood of $x$ and $e^\astp>0$ there exists a function $h \in B_{\mathcal H}$ such that $h(x)=1$ and $\abs{h}<e^\astp$ on $K u^\astetminus U$).

\begin{thm}
	\langlebel{jednostranna}
	For $i=1, 2$, let $\mathcal H_i$ be a closed subspace of $\mathcal C(K_i, E_i)$, where $K_i$ is a compact Hausdorff space and $E_i$ is a Banach space. Let $E_2$ do not contain an isomorphic copy of $c_0$, and let $\mathcal B$ be a closed boundary for $\mathcal H_2$. Suppose that there exists an into isomorphism $T:\mathcal H_1 |ightarrow \mathcal H_2$ with $\norm{T}\norm{T^{-1}}<3$. Then:
	\begin{itemize}
		\item[a)] If $\alpha$ is an ordinal such that $\mathcal W_{\mathcal H_1}^{(\alpha)}$ is nonempty, then so is $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$.
		\item[b)] If $\alpha$ is an ordinal such that $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is finite, then so is $\mathcal W_{\mathcal H_1}^{(\alpha)}$.
		\item[c)] Let $\alpha$ be an ordinal such that $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is finite. Then $E_2^m$ contains an isomorphic copy of $E_1^n$, where $\abs{\mathcal W_{\mathcal H_1}^{(\alpha)}}=n$, $\abs{\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}}=m$. In particular, if $E_1=E_2$ is a finite-dimensional space, then $n \leq m$.
		\item[d)] Let $\mathcal H_1$ contains the constant functions. Then for an ordinal $\alpha$ such that $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is infinite it holds that 
		$\abs{\mathcal W_{\mathcal H_1}^{(\alpha)}} \leq \abs{\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}}$.
	e^\astnd{itemize}
e^\astnd{thm}

We point out that in Theorem |ef{jednostranna}, the assumption of weak peak points is imposed only on one of the two considered function spaces. This may be surprising, as it is known that the Amir-Cambern theorem for subspaces fails completely, if the Choquet boundary of at least one of the two function spaces does not consist of weak peak points. Indeed, in \cite{hess}, Hess shows that for each $e^\astp>0$, there exists a space $\mathcal H_2 u^\astubseteq \mathcal C([0, \omega])$ with $\mathcal Ch_{\mathcal H_2} K_2=[0, \omega)$, such that there exists an isomorphism $T: \mathcal H_1=\mathcal C([0, \omega]) |ightarrow \mathcal H_2$ with $\norm{T}\norm{T^{-1}}<1+e^\astp$. Here, of course, the space $\mathcal H_2$ does not satisfy the assumption of weak peak points. The reason why this example does not contradict Theorem |ef{jednostranna} is that in this case $\Omega_{\mathcal H_2}=[0, \omega]$, and the smallest closed boundary for $\mathcal H_2$ is $\mathcal B=\overline{\mathcal Ch_{\mathcal H_2} K_2}=[0, \omega]$, thus
\[\mathcal W_{\mathcal H_1}=[0, \omega]=\mathcal B \cap \Omega_{\mathcal H_2}.\]
Theorem |ef{jednostranna} shows that if the topological differences of the set of weak peak points of one space and a boundary of the other space are more significant than in this case, then such a thing is no longer possible. This example also shows that in Theorem |ef{jednostranna}, we cannot omit the assumption that the boundary is closed.

The following theorem extends \cite[Theorem 1.5]{CandidoGalego3}.

\begin{thm}
	\langlebel{homeomorfni}
	For $i=1, 2$, let $K_i$ is a compact Hausdorff space, $E_i$ is a Banach space not containing an isomorphic copy of $c_0$, and $\mathcal H_i$ is a closed subspace of $\mathcal C(K_i, E_i)$ such that $\mathcal Ch_{\mathcal H_i} K_i$ is closed and countable, and consists of weak peak points. Let for each natural numbers $m<n$, $E_1^m$ does not contain an isomorphic copy of $E_2^n$, and $E_2^m$ does not contain an isomorphic copy of $E_1^n$. If there exists an isomorphism $T: \mathcal H_1 |ightarrow \mathcal H_2$ with $\norm{T}\norm{T^{-1}}<3$, then $\mathcal Ch_{\mathcal H_1} K_1$ is homeomorphic to $\mathcal Ch_{\mathcal H_2} K_2$.
e^\astnd{thm}

\begin{proof}
	Since $\mathcal Ch_{\mathcal H_1} K_1$ and $\mathcal Ch_{\mathcal H_2} K_2$ are countable compact Hausdorff spaces, there exist nonzero ordinals $\alpha, \beta$ and $m, n \in e^\astn$ such that 
	$\mathcal Ch_{\mathcal H_1} K_1$ si homeomorphic to $[1, \omega^{\alpha}m]$ and $\mathcal Ch_{\mathcal H_2} K_2$ is homeomorphic to $[1, \omega^{\beta}n]$, see \cite[Proposition 8.6.5]{semadeni}. By using Theorem |ef{jednostranna} a) with $\mathcal W_{\mathcal H_1}=\mathcal Ch_{\mathcal H_1} K_1$, $\mathcal B=\mathcal Ch_{\mathcal H_2} K_2u^\astubseteq \Omega_{\mathcal H_2}$, we obtain that $\alpha \leq \beta$. Thus $\alpha=\beta$ by symmetry. By using c) of Theorem |ef{jednostranna} twice it follows that also $m$ is equal to $n$, which concludes the proof. 
e^\astnd{proof}

If $\mathcal H$ is an extremely regular subspace of $\mathcal C(K)$, then it is clear that each point of $K$ is a weak peak point with respect to $\mathcal H$, and thus $\mathcal Ch_{\mathcal H} K=K$, see \cite[Lemma 2.5]{rondos-spurny-vector-spaces}. Thus extremely regular spaces serve as a simple example of spaces that have closed boundaries consisting of weak peak points.

It is natural to ask whether the assumption that the boundaries are closed is necessary in Theorem |ef{homeomorfni} (we remind that in the case of the Amir-Cambern theorem for subspaces, no topological assumptions need to be imposed on the boundaries, see \cite[Theorem 1.1]{rondos-spurny-spaces} or \cite[Theorem 1.1]{rondos-spurny-vector-spaces}). In section 3, we present a simple example showing that Theorem |ef{homeomorfni} does not hold without the assumption of closed Choquet boundaries (see Example |ef{Priklad}). 

Next we turn our attention to isomorphisms that are not necessarily bounded by the number $3$. It is well known that if two $\mathcal C(K)$ spaces are isomorphic, then the underlying compact spaces have the same cardinality, see \cite{cengiz} for the case of scalar functions and \cite{GalegoVillamizar} for an analogous result for vector-valued functions. Generalizations of those results to the context of subspaces were given in \cite{rondos-spurny-spaces} and \cite{rondos-spurny-vector-spaces}. 

Also, it is known that isomorphisms between $\mathcal C(K)$ spaces are somehow connected with the scattered structures of the underlying compact spaces. Indeed, it was proved in \cite[Theorem 1.4]{CandidoScattered} that if $E$ is a Banach space not containing an isomorphic copy of $c_0$, $K_2$ is a scattered locally compact space and $\mathcal C_0(K_1)$ embeds isomorphically into $\mathcal C_0(K_2, E)$, then $K_1$ is also scattered. 

Moreover, there have been proven estimates of the Banach-Mazur distance of $\mathcal C(K)$ spaces from $\mathcal C_0(\mathcal Gamma, E)$ spaces, where $\mathcal Gamma$ is a discrete set, and from $\mathcal C(F)$, where $F$ is a compact space of height $2$ (in particular for $F=[0, \omega]$), based on the height of the compact space $K$. 
It was proved in \cite[Theorem 1.2]{CandidoGalegoComega} that if $K$ is a compact space with $K^{(n)} \neq e^\astmptyset$ for some $n \in e^\astn$, $F$ is a compact space with $F^{(2)}=e^\astmptyset$ and there exists an isomorphism $T: \mathcal C(K) |ightarrow \mathcal C(F)$, then $\norm{T}\norm{T^{-1}} \geq 2n-1$.  Moreover, if $\abs{K^{(n)}}>\abs{F^{(1)}}$, then $\norm{T}\norm{T^{-1}} \geq 2n+1$. In \cite[Theorem 1.1]{CANDIDOc0} it has been showed that if $\mathcal Gamma$ is an infinite discrete space, $E$ is a Banach space not containing an isomorphic copy of $c_0$ and $T: \mathcal C(K) |ightarrow \mathcal C_0(\mathcal Gamma, E)$ is an into isomorphism, then for each $n \in e^\astn$, if $K^{(n)} \neq e^\astmptyset$, then $\norm{T}\norm{T^{-1}} \geq 2n+1$. 
Similar results for isomorphisms with range in $\mathcal C_0(\mathcal Gamma, E)$ spaces were proven before in \cite{CandidoGalegoFund} and \cite{CandidoGalegoCotype}. In the following theorem, which generalizes the above results, we show that the estimate $3$ for the Banach-Mazur distance of two function spaces obtained in Theorem |ef{jednostranna} may be improved, if the heights of the set of weak peak points and of the closed boundary are finite and the height of the set of weak peak points is significantly larger.

\begin{thm}
	\langlebel{distance}
	For $i=1, 2$, let $K_i$ be a compact space, $E_i$ be a Banach space and $\mathcal H_i$ be a closed subspace of $\mathcal C(K_i, E_i)$. Let $E_2$ do not contain an isomorphic copy of $c_0$, $\mathcal B$ be a closed boundary for $\mathcal H_2$ and $T: \mathcal H_1 |ightarrow \mathcal H_2$ be an into isomorphism. Let $l, k \in e^\astn$, $l \geq k$. Let one of the following conditions hold.
	\begin{itemize}
		\item[(i)]  $\mathcal W_{\mathcal H_1}^{(l)}$ is nonempty and $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is empty. 
		\item[(ii)] $\mathcal W_{\mathcal H_1}^{(l)}$ is infinite and $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is finite.
		\item[(iii)]  $\mathcal W_{\mathcal H_1}^{(l)}$ and $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ are both finite, and $E_2^m$ does not contain an isomorphic copy of $E_1^n$, where $\abs{\mathcal W_{\mathcal H_1}^{(l)}}=n$, $\abs{\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}}=m$. 	
	e^\astnd{itemize}
	Then $\norm{T}\norm{T^{-1}} \geq \max\{3, Fr\'echet\ ac{2l+2-k}{k}\}$.
e^\astnd{thm}

We show that the lower bounds that we obtain here are the same as in \cite{CandidoGalegoComega} and \cite{CANDIDOc0}. Thus let $K_1$ be a compact space with $K^{(n)}$ nonempty and $\mathcal H_1=\mathcal C(K_1)$. Then $\mathcal W_{\mathcal H_1}=K_1$ by the Urysohn Lemma. If $\mathcal H_2=\mathcal C(F)$, where $F^{(2)}=e^\astmptyset$, then let $\mathcal B=F$. Since $K^{(n-1)}$ is infinite and $\mathcal B^{(1)}$ is finite, it follows that from (ii) with $l=n-1$ and $k=1$ that we obtain the same lower bound as in \cite[Theorem 1.2]{CandidoGalegoComega}. From (iii) with $E_1=E_2=e^\astr$, $l=n$ and $k=1$, we also obtain the same bound in the case when $\abs{K^{(n)}}>\abs{F^{(1)}}$. Next, let $\mathcal H_2=\mathcal C_0(\mathcal Gamma, E)$, where $\mathcal Gamma$ is a discrete space. Let $K_2=\mathcal Gamma \cup \{\alpha\}$ be the one-point compactification of $\mathcal Gamma$. Then we may put $\mathcal B=K_2$, $k=1$, $l=n$, and then $\mathcal B^{(1)} \cap \Omega_{\mathcal H_2}=\{\alpha\} \cap \Omega_{\mathcal H_2}=e^\astmptyset$, and from (i) we obtain the same estimate as in \cite[Theorem 1.1]{CANDIDOc0}.

From the fact that the term $Fr\'echet\ ac{2l+2-k}{k}$ appearing in Theorem |ef{distance} can happen to be arbitrary close to $1$, in contrast with the number $3$, it is clear that this estimate is in general far from being optimal. Nevertheless, the estimate might be reasonable in cases when $k$ is substantially smaller than $n$. For $k=1$ it is optimal, since the number is attained e.g. for $\mathcal H_1=\mathcal C([0, \omega^n], E)$, $\mathcal H_2=\mathcal C_0(\omega, E)$, where $E$ is a Banach space not containing an isomorphic copy of $c_0$, see \cite[Theorem 1.2]{CANDIDOc0}. Moreover, Theorem |ef{distance} has the following simple, but interesting consequence.

\begin{cor}
\langlebel{finitefinite}
	For $i=1, 2$, let $K_i$ be a compact space, $E_i$ be a Banach space and $\mathcal H_i$ be a closed subspace of $\mathcal C(K_i, E_i)$. Let $E_2$ do not contain an isomorphic copy of $c_0$, $\mathcal B$ be a closed boundary for $\mathcal H_2$, and suppose that $\mathcal H_1$ is isomorphic to a subspace of $\mathcal H_2$. Then if $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is empty for some $k \in e^\astn$, then $ht(\mathcal W_{\mathcal H_1})$ is finite.
e^\astnd{cor}

\begin{proof}
	Let $T: \mathcal H_1 |ightarrow \mathcal H_2$ be an into isomorphism. Then by Theorem |ef{distance}, $\mathcal W_{\mathcal H_1}^{(l)}$ is empty for \[l>Fr\'echet\ ac{\norm{T}\norm{T^{-1}}k+k-2}{2}.\] Thus $ht(\mathcal W_{\mathcal H_1})$ is finite.	
e^\astnd{proof}

Notice that all the above results hold automatically also for subspaces $\mathcal H u^\astubseteq \mathcal C_0(K, E)$, where $K$ is locally compact. Indeed, by \cite[Lemma 2.10]{rondos-spurny-vector-spaces}, if $J=K \cup \{\alpha\}$ is the one-point compactification of $K$, then $\mathcal H$ is isometric to a subspace $\wedgeidetilde{\mathcal H}$ of $J$, defined by
	\[
\wedgeidetilde{\mathcal H}=\{h\in \mathcal C(J,E)\operatorname{co}lon h|_{K}\in \mathcal H\ \&\ h(\alpha)=0\}.
\]
Moreover, $\mathcal Ch_{\mathcal H} K$ is homeomorphic to $\mathcal Ch_{\wedgeidetilde{\mathcal H}} J$ and a point $x \in \mathcal Ch_{\mathcal H} K$ is a weak peak point with respect to $\mathcal H$ if and only if it is a weak peak point with respect to $\wedgeidetilde{\mathcal H}$. Also, it is clear that if $\mathcal B$ is a closed boundary for $\mathcal H$, then $\wedgeidetilde{\mathcal B}=\mathcal B \cup \{\alpha\}$ is a closed boundary for $\wedgeidetilde{\mathcal H}$, and $\wedgeidetilde{\mathcal B} \cap \Omega_{\wedgeidetilde{\mathcal H}}=(\mathcal B \cup \{\alpha\}) \cap \Omega_{\wedgeidetilde{\mathcal H}}=\mathcal B \cap \Omega_{\mathcal H}$. Thus the results valid for $\wedgeidetilde{\mathcal H}$ apply also on $\mathcal H$.

Next, we point out what the above results give for the case of $\mathcal C(K, E)$ spaces.

\begin{cor}
\langlebel{Ckacka}
Let $K_1, K_2$ be infinite compact spaces, $E_1, E_2$ be Banach spaces, $E_2$ not containing an isomorphic copy of $c_0$. 
\begin{itemize}
	\item[(i)] If $l, k \in e^\astn$, $l>k$, $ht(K_1)=l$, $ht(K_2)=k$, and there exists an into isomorphism $T:\mathcal C(K_1, E_1) |ightarrow \mathcal C(K_2, E_2)$, then \[\norm{T}\norm{T^{-1}} \geq \max\{3, Fr\'echet\ ac{2l-k-1}{k-1}\}.\]
	\item[(ii)]
	If $\mathcal C(K_1, E_1)$ embeds isomorphically into $\mathcal C(K_2, E_2)$ and $ht(K_2)$ is finite, then $ht(K_1)$ is finite.
e^\astnd{itemize} 
e^\astnd{cor}

\begin{proof}
By the Urysohn Lemma, it is clear that each point of $K_1$ is a weak peak point with respect to $\mathcal C(K_1, E_1)$, and consequently, $\mathcal Ch_{\mathcal H_1} K_1=K_1$, see \cite[Lemma 2.5]{rondos-spurny-vector-spaces}. We denote $\mathcal B=K_2$. Since $K_2^{(k-1)}$ is finite, and $K_1^{(l-2)}$ is infinite, with the use of Theorem |ef{distance}(ii) we obtain the estimates in the statement (i) (the bound $3$ was known before, see \cite[Theorem 2.1]{CandidoGalego3}). The statement (ii) follows in the same way as Corollary |ef{finitefinite}.
e^\astnd{proof}

u^\astection{Auxiliary results}

This section contains the unexplained definitions and notations, as well as several simple results that will be frequently used later.

To start with, the cardinality of a set $M$ is denoted by $\abs{M}$. Next, the e^\astmph{derivative} of a topological space $S$ is defined recursively as follows. The set $S^{(1)}$ is the set of accumulation points of $S$, and for an ordinal $\alpha>1$, let $S^{(\alpha)}=(S^{\beta})^{(1)}$, if $\alpha=\beta+1$, and $S^{(\alpha)}=\bigcap_{\beta<\alpha} S^{(\beta)}$, if $\alpha$ is a limit ordinal. Moreover, let $S^{(0)}=S$. The topological space $S$ is called e^\astmph{scattered} if there exists an ordinal $\alpha$ such that $S^{(\alpha)}$ is empty, and minimal such $\alpha$ is called the e^\astmph{height} of $S$ and is denoted by $ht(S)$. Thus if $K$ is a scattered compact space, then $ht(K)$ is a successor ordinal, $K^{(ht(K)-1)}$ is finite, and $K^{(\alpha)}$ is infinite for each ordinal $\alpha<ht(K)-1$. All topological spaces are assumed to be Hausdorff.

All Banach spaces are tacitly assumed to be real and of dimension at least $1$. If $E$ is a Banach space then $E^*$ stands for its dual space. We denote by $B_E$ and $S_E$ the unit ball and sphere in $E$, respectively, and we write $\langle\cdot,\cdot|a\operatorname{co}lon E^*v^\astimes Ev^\astoe^\astr$ for the duality mapping. All isomorphisms between Banach spaces are assumed to be surjective, otherwise they are referred to as into isomorphisms. The e^\astmph{Banach-Mazur distance} of Banach spaces $E_1$, $E_2$ is defined to be the infimum of $\norm{T}\norm{T^{-1}}$ over the set of all isomorphisms $T:E_1 |ightarrow E_2$ and is denoted by $d_{BM}(E_1, E_2)$. 

Let $K$ be a locally compact space and $E$ be a Banach space. For $h \in \mathcal C_0(K, E)$ and $e^\ast \in E^\ast$, $e^\ast(h)$ is the element of $\mathcal C_0(K)$ defined by $e^\ast(h)(x)=\langle e^\ast, h(x) |a$ for $x \in K$. Next, for a function $f \in \mathcal C_0(K)$ and $e \in E$, the function $f \otimes e \in \mathcal C_0(K, E)$ is defined by
\[ (f \otimes e)(x)=f(x)e, \quad x \in K .\]

If $\mathcal H$ is a closed subspace of $\mathcal C_0(K, E)$, then its canonical scalar function space $\mathcal A u^\astubseteq \mathcal C_0(K)$ is defined as the closed linear span of the set 
\[ \lbrace e^\ast(h): e^\ast \in E^\ast, h \in \mathcal H |brace  u^\astubseteq \mathcal C_0(K).\]
Moreover, let $\mathcal A^E$ stands for the subspace of $\mathcal A$ consisting of all functions $h$ from $\mathcal A$ satisfying that $h \otimes e \in \mathcal H$ for each $e \in E$. The e^\astmph{Choquet boundary} $\mathcal Ch_{\mathcal H} K$ of $\mathcal H$ is defined as the Choquet boundary of $\mathcal A$, that is, $\mathcal Ch_{\mathcal H} K$ is the set of those points $x \in K$ such that the functional 
\[i(x): h \mapsto h(x), \quad h \in \mathcal A\] 
is an extreme point of $B_{\mathcal A^{*}}$. It follows by \cite[Lemma 2.1]{rondos-spurny-vector-spaces} and \cite[Theorem 2.3.8]{fleming-jamison-1} that the Choquet boundary is a boundary for $\mathcal H$, that is,
\[\norm{h}=u^\astup_{x \in \mathcal Ch_{\mathcal H} K} \norm{h(x)}, \quad h \in \mathcal H.\]
Thus a canonical example of a closed boundary for $\mathcal H$ is the closure $\overline{\mathcal Ch_{\mathcal H} K}$ of the Choquet boundary of $\mathcal H$. Moreover, the following strict maximum principle holds. If $h$ is a function from $\mathcal A$ and $C \in e^\astr$, then $h<C$ on $K$ provided $h<C$ on $\mathcal Ch_{\mathcal H} K$, see \cite[Proposition 3.87]{lmns}. Also, it is easy to see that the Choquet boundary $\mathcal Ch_{\mathcal H} K$ is a subset of $\Omega_{\mathcal H}$. Indeed, the space $\mathcal A$ is nontrivial by our assumption, and hence $0$ is not an extreme point of $\mathcal B_{\mathcal A^*}$.

In this paper, we use a slightly stronger definition of weak peak points for vector-valued subspaces than in \cite{rondos-spurny-vector-spaces}. Here we say that a point $x \in K$ is a e^\astmph{weak peak point} (with respect to $\mathcal H$), if for each neighbourhood $U$ of $x$ and $e^\astp \in (0, 1)$ there exists a function $h \in \mathcal A^E$, such that 
\[0 \leq h \leq 1, h(x)>1-e^\astp v^\astext{ and } h<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H} K u^\astetminus U.\] 
Thus if $\mathcal H=\mathcal C(K, E)$ then $\mathcal A^E=\mathcal A=\mathcal C(K)$, and each point of $K$ is a weak peak point. If $X$ is a compact convex set in a locally convex space and $\mathcal H=Fr\'echet\ a(X, E)$, the space of affine $E$-valued continuous functions on $X$, then $\mathcal A^E=\mathcal A=Fr\'echet\ a(X, e^\astr)$, and a point $x \in X$ is a weak peak point with respect to $\mathcal H$ if and only if it is a weak peak point in the classical sense. We also note that the set $\mathcal W_{\mathcal H}$ of weak peak points of $\mathcal H$ is a subset of the Choquet boundary of $\mathcal H$, see \cite[Lemma 2.5]{rondos-spurny-vector-spaces}.

For the proof of the following Lemma, we recall that a series $u^\astum_{i=1}^{\infty} e_i$ in a Banach space $E$ is e^\astmph{weakly unconditionally Cauchy} if $u^\astum_{i=1}^{\infty} \abs{\langle e^\ast, e_i |a}< \infty$ for each $e^\ast \in E^\ast$.

\begin{lemma}
	\langlebel{c_0}
	Let for $i=1, 2$, $K_i$ is a compact space, $E_i$ is a Banach space and $\mathcal H_i$ is a closed subspace of $\mathcal C(K_i, E_i)$. Let $\mathcal A_1$ be the scalar function space associated to $\mathcal H_1$, let $T:\mathcal H_1 |ightarrow \mathcal H_2$ be an into isomorphism, and fix a nonzero vector $e \in E_1$. Suppose that $\{ f_i \}$ is a sequence in $\mathcal A_1$ satisfying that $h_i=f_i \otimes e \in \mathcal H_1$ for each $i \in e^\astn$. Suppose that there exists a constant $C \in e^\astr$ satisfying that for each $n \in e^\astn$ and $\alpha_1, \ldots , \alpha_n \in S_{e^\astr}$ it holds that
	\[\norm{u^\astum_{i=1}^n \alpha_i h_i}<C.\]
	Moreover, let $y \in K_2$ and $e^\astp>0$ satisfy that 
	$\norm{Th_i(y)}>e^\astp$ for each $i \in e^\astn$. Then $E_2$ contains an isomorphic copy of $c_0$.
e^\astnd{lemma}

\begin{proof}
	We consider the evaluation mapping $\phi\operatorname{co}lon K_2 v^\astimes E^\ast_2 v^\asto \mathcal H_2^*$ defined as
	\[ \langle \phi(y, e^\ast), g |a=\langle e^\ast, g(y) |a, \quad g \in \mathcal H_2, y \in K_2, e^\ast \in E^\ast_2.\] 
	By the classical characterization of the Banach spaces containing $c_0$, see \cite[Theorem 6.7]{morrison2001functional}, it is enough to show that the series $u^\astum_{i=1}^{\infty} Th_i(y)$ is weakly unconditionally Cauchy in $E_2$. Thus fix $e^* \in S_{E_2^*}$, and let $T^*$ be the adjoint of $T$. 
	Fix $n \in e^\astn$, and let $\alpha_1, \ldots, \alpha_n \in S_{e^\astr}$ satisfy 
	\[\abs{\langle T^*\phi(y, e^*), h_i |a}=\alpha_i \langle T^*\phi(y, e^*), h_i |a, \quad i=1, \ldots, n.\]
	Then we have 
	\begin{equation}
	\nonumber
	\begin{aligned}
	&u^\astum_{i=1}^{n} \abs{\langle e^*, Th_i(y) |a}=u^\astum_{i=1}^{n} \abs{\langle \phi(y, e^*), Th_i |a}=
	u^\astum_{i=1}^{n} \abs{\langle T^*\phi(y, e^*), h_i |a}
	=\\&=
	u^\astum_{i=1}^{n} \alpha_i \langle T^*\phi(y, e^*), h_i |a=\langle T^*\phi(y, e^*), u^\astum_{i=1}^{n} \alpha_i h_i |a 
	\leq \\& \leq
	\norm{T^*\phi(y, e^*)} \norm{u^\astum_{i=1}^n \alpha_i h_i}\leq \norm{T^*} \norm{e^*}{\norm{u^\astum_{i=1}^n \alpha_i h_i}} < C \norm{T^*}. 
	e^\astnd{aligned}
	e^\astnd{equation}
	Thus also $u^\astum_{i=1}^{\infty} \abs{\langle e^*, Th_i(y) |a} \leq C\norm{T^*}<\infty$, which finishes the proof.
e^\astnd{proof}

We also prove the following two simple observations, as we will be using them repeatedly later on.

\begin{lemma}
\langlebel{omezenost}
Let $K$ be a compact space, $E$ be a Banach space and $\mathcal H$ be a closed subspace of $\mathcal C(K, E)$. Let $n \in e^\astn$, $C \in e^\astr$, $\{\alpha_i\}_{i=1}^n$ be positive numbers, $\{U_i\}_{i=1}^n$ be pairwise disjoint nonempty open sets in $K$, and let functions $\{h_i\}_{i=1}^n u^\astubseteq \mathcal H$ satisfy that for each $i=1 \ldots, n$, $\norm{h_i} \leq C$ on $K$ and $\norm{h_i}<\alpha_i$ on $\mathcal Ch_{\mathcal H} K u^\astetminus U_i$. Then
\[\norm{u^\astum_{i=1}^n h_i} \leq C+u^\astum_{i=1}^n \alpha_i.\]
e^\astnd{lemma}

\begin{proof}
If $x \in U_{i_0} \cap \mathcal Ch_{\mathcal H} K$ for some $i_0 \in \{1, \ldots, n\}$, then since the sets $U_i$ are pairwise disjoint, 
\[\norm{u^\astum_{i=1}^n h_i(x)} \leq u^\astum_{i=1}^n \norm{h_i(x)} \leq C+u^\astum_{i \neq \i_0} \alpha_i.\]
If $x \in \mathcal Ch_{\mathcal H} K u^\astetminus \bigcup_{i=1}^n U_i$, then 
\[\norm{u^\astum_{i=1}^n h_i(x)} \leq u^\astum_{i=1}^n \norm{h_i(x)} \leq u^\astum_{i=1}^n \alpha_i.\]
Thus $\norm{u^\astum_{i=1}^n h_i(x)} \leq C+u^\astum_{i=1}^n \alpha_i$ for $x \in \mathcal Ch_{\mathcal H} K$, and hence $\norm{u^\astum_{i=1}^n h_i(x)} \leq C+u^\astum_{i=1}^n \alpha_i$ for $x \in K$.
e^\astnd{proof}

\begin{lemma}
	\langlebel{nerovnost}
	Let $K$ be a compact space, $\mathcal A u^\astubseteq \mathcal C(K)$ be a closed subspace, $0<e^\astp<1$, and functions $h, g \in \mathcal A$ satisfy that $0 \leq g \leq 1, 0 \leq h \leq 1$ on $K$, $h>1-e^\astp$ on an open set $U$ and $g<e^\astp$ on $\mathcal Ch_{\mathcal A} K u^\astetminus U$. Then $g<h+e^\astp$ on $K$.
e^\astnd{lemma}

\begin{proof}
	If $x \in \mathcal Ch_{\mathcal A} K \cap U$, then \[g(x) \leq 1=1-e^\astp+e^\astp<h(x)+e^\astp.\] On the other hand, if $x \in \mathcal Ch_{\mathcal A} K u^\astetminus U$, then	
	\[g(x)<e^\astp \leq h(x)+e^\astp.\]
	Thus $g-h<e^\astp$ on $\mathcal Ch_{\mathcal A} K$, and hence $g-h<e^\astp$ on $K$.
e^\astnd{proof}

u^\astection{Isomorphisms with bound 3}

Throughout this section, let for $i=1, 2$, $K_i$ be a compact space, $E_i$ be a Banach space and $\mathcal H_i$ be a closed subspace of $\mathcal C(K_i, E_i)$. Let $\mathcal A_1$ be the scalar function space associated to $\mathcal H_1$ and $\mathcal B$ be a closed boundary for $\mathcal H_2$.
For $g \in \mathcal H_2$ and $e^\astp>0$ we denote 
\[\mathcal R(g, e^\astp)=\{ y \in K_2 \operatorname{co}lon \norm{g(y)} \geq e^\astp\}.\]
Notice that from the definition it is clear that for each function $g \in \mathcal H_2$, $\mathcal R(g, e^\astp) u^\astubseteq \Omega_{\mathcal H_2}$.
Next, inspired by \cite{CandidoGalego3}, we define certain sets $\mathcal F$, $\mathcal Lambda$, which we use for the proof of Theorem |ef{jednostranna}. For $0 < e^\astp <1$, functions $f_1, \ldots, f_n \in \mathcal A_1^{E_1}$ and points $z_1, \ldots, z_n \in \mathcal Ch_{\mathcal H_1} K_1$ we denote 
\begin{equation}
\nonumber
\begin{aligned}
\mathcal{F}((f_i, z_i)_{i=1}^n, e^\astp)
=&\{(g_1, \ldots, g_n) \in (\mathcal A_1^{E_1})^n \forall i=1, \ldots, n : 0 \leq g_i \leq 1, \\& g_i(z_i)>1-e^\astp v^\astext{ and } g_i <f_i+e^\astp\}.
e^\astnd{aligned}
e^\astnd{equation}

Suppose moreover that $T:\mathcal H_1 |ightarrow \mathcal H_2$ is an into isomorphim, $e_1, \ldots, e_n \in E_1$ are nonzero vectors with $\max_{i=1, \ldots , n} \norm{e_i}=1$, $U_1, \ldots, U_n$ are pairwise disjoint nonempty open sets in $K_1$, and let $h_1, \ldots, h_n$ be functions in $\mathcal A_1^{E_1}$ satisfying that 
\[0 \leq h_i \leq 1, \norm{h_i}>1-e^\astp v^\astext{ and } h_i<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i, \quad i=1, \ldots, n.\]
Then for points $x_1, \ldots, x_n \in \mathcal Ch_{\mathcal H_1} K_1$ satisfying that $h_i(x_i)>1-e^\astp$ for each $i=1, \ldots, n$, we denote
\begin{equation}
\nonumber
\begin{aligned}
\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)}((x_i)_{i=1}^n)
=\mathcal B \cap \bigcap_{(g_1, \ldots, g_n) \in \mathcal{F}((h_i, x_i)_{i=1}^n, e^\astp)} \mathcal R(T(u^\astum_{i=1}^n g_i \otimes e_i), e^\astp).
e^\astnd{aligned}
e^\astnd{equation}

Notice that the set $\mathcal{F}((h_i, x_i)_{i=1}^n, e^\astp)$ is nonempty, as it contains $(h_1, \ldots, h_n)$. The following technical Lemma, describing the properties of the above defined set $\mathcal Lambda$, contains the main part of the proof of Theorem |ef{jednostranna}.

\begin{lemma}
	\langlebel{main}
	Let $E_2$ do not contain an isomorphic copy of $c_0$. Let $n \in e^\astn$, and suppose that there exists an into  isomorphism $T: \mathcal H_1 |ightarrow \mathcal H_2$ with $\norm{T}<3$ and $\norm{T^{-1}}=1$. Fix	
	 \[0<e^\astp<Fr\'echet\ ac{3-\norm{T}}{\norm{T}(7n+2)+3n+2}.\]
	 Moreover, let $e_1, \ldots, e_n \in E_1$ be nonzero vectors with $\max_{i=1, \ldots , n} \norm{e_i}=1$, $U_1, \ldots, U_n$ be pairwise disjoint nonempty open subsets of $K_1$, and let $h_1, \ldots, h_n$ be functions in $\mathcal A_1^{E_1}$ satisfying that 
	 \[0 \leq h_i \leq 1, \norm{h_i}>1-e^\astp v^\astext{ and } h_i<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i, \quad i=1, \ldots, n.\]
	
	Then:
	\begin{itemize}
		\item [a)] If for $i=1, \ldots, n$, $\{x_i^k\}_{k \in e^\astn} u^\astubseteq \mathcal W_{\mathcal H_1}$ are infinite sequences of pairwise distinct points satisfying that 
		\[h_i(x_i^k)>1-e^\astp, \quad i=1, \ldots, n, k \in e^\astn,\]
		 then the intersection $\bigcap_{k \in e^\astn} \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i^k)_{i=1}^n)$ is empty.
		
		\item[b)] If $x_1, \ldots, x_n \in \mathcal W_{\mathcal H_1}^{(\alpha)}$ for an ordinal $\alpha$ are points satisfying that
		\[h_i(x_i)>1-e^\astp, \quad i=1, \ldots, n,\]
		then $\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)}((x_i)_{i=1}^n) \cap \mathcal B^{(\alpha)} \neq e^\astmptyset$.
	e^\astnd{itemize}
e^\astnd{lemma}

\begin{proof}
	For the proof of a), suppose that for $i=1, \ldots, n$ there exit such sequences $\{x_i^k\}_{k \in e^\astn}$ and $y \in \bigcap_{k \in e^\astn} \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i^k)_{i=1}^n)$. Passing to subsequences and using the continuity of the functions $h_i$, we may assume that there exist pairwise disjoint open sets $\{V_i^k\}_{k \in e^\astn}, i=1, \ldots, n$, each containing $x_i^k$, and such that $h_i>1-e^\astp$ on $V_i^k$. For each $i$ and $k$, since $x_i^k$ is a weak peak point, we may find a function $g_i^k \in \mathcal A_1^{E_1}$ satisfying that
	\[0 \leq g_i^k \leq 1, g_i^k(x_i^k)>1-Fr\'echet\ ac{e^\astp}{2^k} v^\astext{ and } g_i^k<Fr\'echet\ ac{e^\astp}{2^k} v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus V_i^k.\]
	Notice that for each $i=1, \ldots, n$, since the sets $\{V_i^k\}_{k \in e^\astn}$ are pairwise disjoint, for each $m \in e^\astn$ and $\alpha_1, \ldots , \alpha_m \in S_{e^\astr}$ it holds that $\norm{u^\astum_{k=1}^m \alpha_k g_i^k}\leq 1+e^\astp$
	by Lemma |ef{omezenost}. Also, for each $i, k$, we have $g_i^k<h_i+e^\astp$ (see Lemma |ef{nerovnost}), thus for each $k \in e^\astn$ it holds that $(g_1^k, \ldots, g_n^k) \in \mathcal F((h_i, x_i^k)_{i=1}^n, e^\astp)$. Hence $\norm{T(u^\astum_{i=1}^n (g_i^k \otimes e_i))(y)} \geq e^\astp$ by the definition of $\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)}((x_i^k)_{i=1}^n)$.
	
	Consequently, for each $k \in e^\astn$ there exists $i \in \{1, \ldots, n\}$ such that 
	\[\norm{T(g_i^k \otimes e_i)(y)}\geq Fr\'echet\ ac{e^\astp}{n}.\] Thus there exists $i_0 \in \{1, \ldots, n\}$ such that the set
	\[N=\{ k \in e^\astn: \norm{T(g_{i_0}^k \otimes e_{i_0})(y)}\geq Fr\'echet\ ac{e^\astp}{n}\}\] is infinite. But since $E_2$ does not contain a copy of $c_0$, the sequence $\{g_{i_0}^k\}_{k \in N}$ contradicts Lemma |ef{c_0}, which finishes the proof of a).
	
	For the proof of b) we proceed by transfinite induction. For $\alpha=0$ we want to prove that $\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)}((x_i)_{i=1}^n)$ is a nonempty set for each points $x_1, \ldots, x_n \in \mathcal W_{\mathcal H_1}$ satisfying $h_i(x_i)>1-e^\astp$ for each $i=1, \ldots, n$. Notice that the set  \\$\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)}((x_i)_{i=1}^n)$ is compact, as it is an intersection of closed subsets of $\mathcal B$. Thus it is enough to show that the collection of sets
	\[\{y \in \mathcal B: \norm{T(u^\astum_{i=1}^n g_i \otimes e_i)(y)} \geq e^\astp \}, \quad (g_i)_{i=1}^n \in \mathcal{F}((h_i, x_i)_{i=1}^n, e^\astp) \] 
	has the finite intersection property. So, let functions $g_i^k \in \mathcal A_1^{E_1}$, $i=1, \ldots, n$, $k=1, \ldots, p$ satisfy
	\[0 \leq g_i^k \leq 1, g_i^k(x_i)>1-e^\astp v^\astext{ and } g_i^k <h_i+e^\astp.\]
	By the continuity of the functions $g_i^k$, for each $i=1, \ldots, n$ there exists an open set $V_i u^\astubseteq U_i$ containing $x_i$, and such that $g_i^k>1-e^\astp$ on $V_i$ for each $k=1, \ldots, p$. Since $x_i$ is a weak peak point, there exists a function $f_i \in \mathcal A_1^{E_1}$ satisfying that 
	\[0 \leq f_i \leq 1, f_i(x_i)>1-e^\astp v^\astext{ and } f_i <e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus V_i.\]
	Then $f_i<g_i^k+e^\astp$ for each $i$ and $k$ by Lemma |ef{nerovnost}. 
	
	Next, fix $i_0 \in \{1, \ldots, n\}$ satisfying that $\norm{e_{i_0}}=1$. We have 
	\[\norm{((2f_{i_0}+h_{i_0})\otimes e_{i_0})(x_{i_0})} =\abs{2f_{i_0}(x_{i_0})+h_{i_0}(x_{i_0})}\norm{e_{i_0}}>3-3e^\astp,\]
	and for $i \neq i_0$, 
	\[\norm{((2f_{i}+h_{i})\otimes e_{i})(x_{i_0})} = \abs{2f_{i}(x_{i_0})+h_{i}(x_{i_0})}\norm{e_i}<3e^\astp.\]
	Consequently, and since $\norm{T^{-1}}=1$, we have 
	\begin{equation}
	\nonumber
	\begin{aligned}
	&\norm{T(u^\astum_{i=1}^n (2f_{i}+h_{i})\otimes e_{i})} \geq \norm{u^\astum_{i=1}^n (2f_{i}+h_{i})\otimes e_{i}}
	\geq \norm{(u^\astum_{i=1}^n (2f_{i}+h_{i})\otimes e_{i})(x_{i_0})}
	\geq \\& \geq
	\norm{((2f_{i_0}+h_{i_0})\otimes e_{i_0})(x_{i_0})}-u^\astum_{i \neq i_0} \norm{((2f_{i}+h_{i})\otimes e_{i})(x_{i_0})} > 3-3e^\astp-(n-1)3e^\astp.
	e^\astnd{aligned}
	e^\astnd{equation}
	Hence there exists a point $y \in \mathcal B$ satisfying that
	\[\norm{T(u^\astum_{i=1}^n (2f_{i}+h_{i})\otimes e_{i})(y)}>3-3ne^\astp.\]
	Now, to finish the proof for $\alpha=0$ we need to show that for each $k=1, \ldots, p$,  $\norm{T(u^\astum_{i=1}^n g_i^k \otimes e_i)(y)} \geq e^\astp$. Thus fix $k \in \{1, \ldots, p\}$. Then for each $i=1, \ldots, n$, since $g_i^k<h_i+e^\astp$ and $f_i<g_i^k+e^\astp$, we have 
	\begin{equation}
	\nonumber
	\begin{aligned}
	-1-e^\astp \leq -g_i^k-e^\astp \leq 2f_i-g_i^k-e^\astp \leq 2f_i+h_i-2g_i^k \leq h_i+2e^\astp \leq 1+2e^\astp.
	e^\astnd{aligned}
	e^\astnd{equation} 
	Thus $\norm{(2f_i+h_i-2g_i^k) \otimes e_i} \leq 1+2e^\astp$.
	
	Moreover, for each $i=1, \ldots, n$ we have $g_i^k<h_i+e^\astp<2e^\astp$ on $\mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i$, and $f_i<e^\astp$ on $\mathcal Ch_{\mathcal H_1} K_1 u^\astetminus V_i u^\astupseteq \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i$.
	Consequently, since the sets $U_i$ are pairwise disjoint, and for each $i=1, \ldots, n$ we have
	$\abs{2f_i+h_i-2g_i^k}<7e^\astp$ on $\mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i$, by Lemma |ef{omezenost} it follows that 
	\begin{equation}
	\nonumber
	\begin{aligned}
	&\norm{u^\astum_{i=1}^n (2f_{i}+h_{i}-2g_i^k)\otimes e_{i}} \leq  1+2e^\astp+7ne^\astp.
	e^\astnd{aligned}
	e^\astnd{equation}
	It follows that if $\norm{T(u^\astum_{i=1}^n g_i^k \otimes e_i)(y)}< e^\astp$, we would have
	\begin{equation}
	\nonumber
	\begin{aligned}
	&\norm{T}(1+2e^\astp+7ne^\astp) \geq \norm{T(u^\astum_{i=1}^n(2f_{i}+h_{i}-2g_i^k)\otimes e_{i})} 
	\geq \\& \geq
	\norm{T(u^\astum_{i=1}^n (2f_{i}+h_{i}-2g_i^k)\otimes e_{i})(y)} 
	\geq \\& \geq
	\norm{T(u^\astum_{i=1}^n (2f_{i}+h_{i})\otimes e_{i})(y)}-2\norm{T(u^\astum_{i=1}^n g_i^k\otimes e_{i})(y)} \geq 3-3ne^\astp-2e^\astp.
	e^\astnd{aligned}
	e^\astnd{equation}
	This contradicts the choice of $e^\astp$ and finishes the proof for $\alpha=0$.
	
	Now we assume that the statement b) holds for an ordinal $\alpha \geq 0$ and let $x_1, \ldots, x_n \in \mathcal W_{\mathcal H_1}^{(\alpha+1)}$ be points satisfying that $h_i(x_i)>1-e^\astp$ for $i=1, \ldots, n$. Then for $i=1, \ldots, n$, there exists a net $\{ x^i_{\langlembda_i} \}_{\langlembda_i \in \mathcal Lambda_i}$ of points in $\mathcal W_{\mathcal H_1}^{(\alpha)} \cap U_i$, distinct from $x_i$, converging to $x_i$, and satisfying that $h_i(x^i_{\langlembda_i})>1-e^\astp$ for each $\langlembda_i$. By the assumption, for each $(\langlembda_1, \ldots, \langlembda_n) \in \mathcal Lambda_1 v^\astimes \ldots v^\astimes \mathcal Lambda_n$ there exists a point 
	\[y_{(\langlembda_i)_{i=1}^n} \in \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x^i_{\langlembda_i})_{i=1}^n) \cap \mathcal B^{(\alpha)}.\] Then we may consider $\{ y_{(\langlembda_i)_{i=1}^n} \}$ as a net in $\mathcal B^{(\alpha)}$ with the ordering given by $(\langlembda_i)_{i=1}^n\leq (v^\astilde{\langlembda}_i)_{i=1}^n$ if $\langlembda_i \leq v^\astilde{\langlembda}_i$ for each $i=1, \ldots, n$. Since $\mathcal B^{(\alpha)}$ is compact, passing to a subnet we may assume that the net $\{ y_{(\langlembda_i)_{i=1}^n} \}$ converges to a point $y \in \mathcal B^{(\alpha)}$. We want to show that $y \in \mathcal B^{(\alpha+1)}$. Assuming the contrary, the net $\{ y_{(\langlembda_i)_{i=1}^n} \}$ is essentially constant, thus there exist $\langlembda_1, \ldots, \langlembda_n$ such that $y=y_{(v^\astilde{\langlembda}_i)_{i=1}^n}$ for each $(v^\astilde{\langlembda}_i)_{i=1}^n \geq (\langlembda_i)_{i=1}^n$. But then for each $(v^\astilde{\langlembda}_i)_{i=1}^n \geq (\langlembda_i)_{i=1}^n$ we have 
	\[y=y_{(v^\astilde{\langlembda}_i)_{i=1}^n} \in \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x^i_{v^\astilde{\langlembda}_i})_{i=1}^n),\]
	which contradicts a). Thus $y \in \mathcal B^{(\alpha+1)}$. 
	
	It remains to show that $y \in \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i)_{i=1}^n)$. 
	Thus we choose arbitrary $(g_1, \ldots, g_n) \in \mathcal{F}((h_i, x_i)_{i=1}^n, e^\astp)$. Then for each $i=1, \ldots, n$, $g_i(x_i)>1-e^\astp$ by the definition, thus there exists $\langlembda_i$ such that $g_i(x^i_{v^\astilde{\langlembda}_i})>1-e^\astp$ for each $v^\astilde{\langlembda}_i>\langlembda_i$. This means that
	\[(g_i)_{i=1}^n \in \mathcal{F}((h_i, x^i_{v^\astilde{\langlembda}_i})_{i=1}^n, e^\astp)\] for $(v^\astilde{\langlembda}_i)_{i=1}^n \geq (\langlembda_i)_{i=1}^n$. Thus for $(v^\astilde{\langlembda}_i)_{i=1}^n \geq (\langlembda_i)_{i=1}^n$ we have 
	\[\norm{T(u^\astum_{i=1}^n g_i \otimes e_i)(y_{(v^\astilde{\langlembda}_i)_{i=1}^n})} \geq e^\astp.\] Consequently, $\norm{T(u^\astum_{i=1}^n g_i \otimes e_i)(y)} \geq e^\astp$ by continuity. This means that $y \in \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i)_{i=1}^n)$.
	
	Finally, we assume that $\alpha$ is a limit ordinal and that b) holds for each ordinal $\beta<\alpha$. If $x_1, \ldots, x_n \in \mathcal W_{\mathcal H_1}^{(\alpha)}$, $h_i(x_i)>1-e^\astp$ for each $i$, then $x_1, \ldots, x_n \in \mathcal W_{\mathcal H_1}^{(\beta)}$ for each $\beta<\alpha$. Hence $\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i)_{i=1}^n) \cap \mathcal B^{(\beta)} \neq e^\astmptyset$ for each such $\beta$ by the assumption. Thus 
	\[\mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i)_{i=1}^n) \cap \mathcal B^{(\alpha)}=\bigcap_{\beta<\alpha} \mathcal Lambda_{(T, (h_i, e_i)_{i=1}^n, e^\astp)} ((x_i)_{i=1}^n) \cap \mathcal B^{(\beta)}\]
	is nonempty, as it is a nonincreasing intersection of compact sets. The proof is finished.
e^\astnd{proof}

Now we are ready to prove Theorem |ef{jednostranna}.

e^\astmph{Proof of Theorem |ef{jednostranna}.}

\begin{proof}
	Without loss of generality we may assume that $\norm{T}<3$ and $\norm{T^{-1}}=1$, otherwise we would multiply $T$ by a suitable constant. Let $\mathcal A_1$ be the scalar function space associated to $\mathcal H_1$.
	
	a) Pick $x \in \mathcal W_{\mathcal H_1}^{(\alpha)}$ and 
	\[ 0<e^\astp<Fr\'echet\ ac{3-\norm{T}}{9\norm{T}+5},\]
	and let $e \in E_1$ be an arbitrary vector of norm $1$.  
	Since $x$ is a weak peak point, there exists a function $h \in \mathcal A_1^{E_1}$ satisfying that $0 \leq h \leq 1$ and $h(x)>1-e^\astp$. Then by Lemma |ef{main} b), the set 
	\[\Omega_{\mathcal H_2} \cap \mathcal B^{(\alpha)} u^\astupseteq \mathcal Lambda_{(T, (h, e), e^\astp)}(x) \cap \mathcal B^{(\alpha)}\]
	is nonempty.
	
	b) Let $\alpha$ be an ordinal such that $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is finite. We assume that $\mathcal W_{\mathcal H_1}^{(\alpha)}$ is infinite and seek a contradiction. Then there exist pairwise disjoint open sets $\{U_m \}_{m \in e^\astn}$ in $K_1$ and points $\{ x_m\}_{m \in e^\astn} u^\astubseteq \mathcal W_{\mathcal H_1}^{(\alpha)}$ such that $x_m \in U_m$ for each $m \in e^\astn$. Fix $e \in S_{E_1}$, \[ 0<e^\astp<Fr\'echet\ ac{3-\norm{T}}{9\norm{T}+5},\]
	and find functions $\{g_m\}_{m \in e^\astn}$ from $\mathcal A_1^{E_1}$ satisfying that for each $m \in e^\astn$,
	\[0 \leq g_m \leq 1, g_m(x_m)>1-Fr\'echet\ ac{e^\astp}{2^m} v^\astext{ and } g_m <Fr\'echet\ ac{e^\astp}{2^m} v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_m.\] 
	Then by Lemmas |ef{c_0} and |ef{omezenost}, for each $y \in K_2$ there are at most finitely many functions $g_m$ such that $\norm{T(g_m \otimes e)(y)} \geq e^\astp$. Thus there exists an index $m_0 \in e^\astn$ satisfying that $\norm{T(g_{m_0} \otimes e)(y)}<e^\astp$ for each $y$ from the finite set $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$. This means that 
	\[\mathcal Lambda_{(T, (g_{m_0}, e), e^\astp)}(x_{m_0}) \cap \mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}=\mathcal Lambda_{(T, (g_{m_0}, e), e^\astp)}(x_{m_0}) \cap \mathcal B^{(\alpha)}\]	is empty. But this contradicts Lemma |ef{main} b), which shows that $\mathcal W_{\mathcal H_1}^{(\alpha)}$ is finite.
	
	c) If $\alpha$ is an ordinal such that $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is finite, then we know by b) that $\mathcal W_{\mathcal H_1}^{(\alpha)}$ is also finite. Thus suppose that $\mathcal W_{\mathcal H_1}^{(\alpha)}=\{ x_1, \ldots, x_n\}$ and $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}=\{y_1, \ldots, y_m\}$. We find pairwise disjoint open sets $U_1, \ldots, U_n$ such that $x_i \in U_i$ for each $i=1, \ldots, n$. 
	
	Fix 
	\[ 0<e^\astp<Fr\'echet\ ac{3-\norm{T}}{\norm{T}(7n+2)+3n+2},\]
	and let $h_1, \ldots, h_n$ be functions in $\mathcal A_1^{E_1}$ satisfying that 
	\[0 \leq h_i \leq 1, h_i(x_i)>1-e^\astp v^\astext{ and } h_i<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_i.\]
	
	We define an operator $S:(E_1^n, \norm{\cdot}_{\max}) |ightarrow \mathcal C(\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}, E_2) u^\astimeq (E_2^m, \norm{\cdot}_{\max})$ as
	\[S(e_1, \ldots, e_n)=T(u^\astum_{i=1}^n h_i \otimes e_i)|_{(\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2})}.\] 
	Then $S$ is clearly linear. 
	
	Moreover, let $e_1, \ldots, e_n \in E_1$ be nonzero vectors with $\max_{i=1, \ldots , n} \norm{e_i}=1$. Then 
	\begin{equation}
	\nonumber
	\begin{aligned}
	&\norm{S(e_1, \ldots, e_m)}_{\max} \leq \norm{T(u^\astum_{i=1}^n h_i \otimes e_i)}\leq  \norm{T} \norm{u^\astum_{i=1}^n h_i \otimes e_i}\leq \norm{T}(1+ne^\astp)
	e^\astnd{aligned}
	e^\astnd{equation}
by Lemma |ef{omezenost}. At the same time, by Lemma |ef{main} b), there exists a point $y \in \mathcal B^{(\alpha)}$ such that $\norm{T(u^\astum_{i=1}^n h_i \otimes e_i)(y)} \geq e^\astp$. It follows that $e^\astp\leq \norm{S} \leq \norm{T}(1+ne^\astp)$, and hence $S$ is an isomorphism from $E_1^n$ into $E_2^m$.

	d) If $\mathcal H_1$ contains the constant functions, then so does the space $\mathcal A_1^{E_1}$. Let $h \in \mathcal A_1^{E_1}$ be the constant function $1$, 	
\[0<e^\astp<Fr\'echet\ ac{3-\norm{T}}{9\norm{T}+5},\]
and let $e \in S_{E_1}$. Then again by Lemma |ef{main} b), for each $x_0 \in \mathcal W_{\mathcal H_1}^{(\alpha)}$ there exists $y \in \mathcal Lambda_{(T, (h, e), e^\astp)}(x_0) \cap \mathcal B^{(\alpha)} u^\astubseteq \Omega_{\mathcal H_2} \cap \mathcal B^{(\alpha)}$. Moreover, by Lemma |ef{main} a), each $y \in \mathcal B^{(\alpha)}$ belongs to at most finitely many sets $\mathcal Lambda_{(T, (h, e), e^\astp)}(x)$, where $x \in \mathcal W_{\mathcal H_1}^{(\alpha)}$.  Thus if $\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}$ is infinite, then 
\[\abs{\mathcal W_{\mathcal H_1}^{(\alpha)}} \leq \abs{\omega}\abs{\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}}=\abs{\mathcal B^{(\alpha)} \cap \Omega_{\mathcal H_2}}.\]
e^\astnd{proof}

To finish this section we now show that we cannot omit the assumption that the boundaries are closed in Theorem |ef{homeomorfni}.

\begin{example}
	\langlebel{Priklad}
	Let $K_1=[1, \omega 2]$, $K_2=[1, \omega]$ and
	\[ \mathcal H_1=\{ f \in \mathcal C([1, \omega 2]): f(\omega 2)=0\}, \mathcal H_2=\mathcal C([1, \omega]).\]
	Then each point of the countable Choquet boundaries of $\mathcal H_1$ and $\mathcal H_2$ is a weak peak point, but $d_{BM}(\mathcal H_1, \mathcal H_2)=2$.
e^\astnd{example}

\begin{proof}
	It is clear that $\mathcal W_{\mathcal H_1}=[1, \omega 2)$. Thus $\mathcal Ch_{\mathcal H_1} K_1 u^\astupseteq \mathcal W_{\mathcal H_1}=[1, \omega 2)$. Since zero is not an extreme point of $B_{\mathcal H_1^*}$, we have $\mathcal Ch_{\mathcal H_1} K_1=\mathcal W_{\mathcal H_1}=[1, \omega 2)$. Moreover, it is clear that $\mathcal W_{\mathcal H_2}=\mathcal Ch_{\mathcal H_2} K_2=[1, \omega]$. Thus $d_{BM}(\mathcal H_1, \mathcal H_2) \geq 2$ by \cite[Theorem 1.1]{rondos-spurny-spaces}. 
	
	On the other hand, we define a mapping $T: \mathcal H_1 |ightarrow \mathcal H_2$ by
	\begin{equation}
	\nonumber
	\begin{aligned}
	& Tf(1)=f(1) \\&
	Tf(2m)=f(m)+f(\omega+m), \quad m \in e^\astn, \\&
	Tf(2m+1)=f(m)-f(\omega+m), \quad m \in e^\astn, \\&
	Tf(\omega)=f(\omega).
	e^\astnd{aligned}
	e^\astnd{equation}
	Then it is easy to check that $T$ is an isomorphism from $\mathcal H_1$ onto $\mathcal H_2$ with $S=T^{-1}$ given by
	\begin{equation}
	\nonumber
	\begin{aligned}
	& Sf(1)=f(1) \\&
	Sf(m)=Fr\'echet\ ac{f(2m)+f(2m+1)}{2}, \quad m \in e^\astn, \\&
	Sf(\omega)=f(\omega), \\&
	Sf(\omega+m)=Fr\'echet\ ac{f(2m)-f(2m+1)}{2}, \quad m \in e^\astn.
	e^\astnd{aligned}
	e^\astnd{equation}
	Moreover, it is easy to see that $\norm{T}=2$ and $\norm{S}=1$, which finishes the proof.
e^\astnd{proof}

u^\astection{Isomorphisms between spaces with boundaries of finite height}

In this section, for $i=1, 2$, let $K_i$ be a compact space, $E_i$ be a Banach space and $\mathcal H_i$ be a closed subspace of $\mathcal C(K_i, E_i)$. Let $E_2$ do not contain an isomorphic copy of $c_0$, let $\mathcal A_1$ be the scalar function space associated to $\mathcal H_1$, $\mathcal B$ be a closed boundary for $\mathcal H_2$ and $T:\mathcal H_1 |ightarrow \mathcal H_2$ be an into isomorphism. In this setting, we prove three auxiliary results, which we then use for the proof of Theorem |ef{distance}. 

\begin{lemma}
	\langlebel{finite}
	Let $0 <e^\astp <1$, $\zeta>0$, $e$ be a nonzero vector in $E_1$,  $\{U_n\}$ be a sequence of pairwise disjoint open subsets of $K_1$ and let the sequence of functions $\{ h_n\}$ in $B_{\mathcal A_1}$ satisfies that for each $n \in e^\astn$, $h_n \otimes e \in \mathcal H_1$ and $\abs{h_n}<Fr\'echet\ ac{e^\astp}{2^n}$ on $\mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_n$. Let $G$ be a finite subset of $K_2$. Then there exists an $m \in e^\astn$ such that $\mathcal R(T(h_m \otimes e), \zeta) \cap G=e^\astmptyset$. 
e^\astnd{lemma}

\begin{proof}
	We assume that this is not true and seek a contradiction. Let $G=\{ y_1, \ldots , y_k\}$, and we denote 
	\[\mathcal Lambda_i = \{ n \in e^\astn: \norm{T(h_n \otimes e) (y_i)} \geq \zeta \}, \quad i=1, \ldots, k.\]
	By the assumption, the union $\bigcup_{i=1}^k \mathcal Lambda_i$ contains all natural numbers. Thus we can fix an index $i_0$ such that the set $\mathcal Lambda_{i_0}$ is infinite. Since the sets $U_n$ are pairwise disjoint, by Lemma |ef{omezenost} for each $n \in e^\astn$ and $\alpha_1, \ldots , \alpha_n \in S_{e^\astr}$ it holds that
	\[\norm{u^\astum_{i=1}^n \alpha_i h_i(x)} \leq 1+e^\astp.\]
	But since $E_2$ does not contain an isomorphic copy of $c_0$, the sequence $\{ h_n\}_{n \in \mathcal Lambda_{i_0}}$ contradicts Lemma |ef{c_0}. The proof is finished.
e^\astnd{proof}

The following Lemma, that is motivated by \cite[Proposition 2.3]{CANDIDOc0}, is a key ingredient, which makes use of the assumption that the considered heights are finite. For the proof, notice that if $F u^\astubseteq K_2$ is a compact set, and the set $F \cap \mathcal B^{(k)}$ is infinite for some $k \in e^\astn$, then $F \cap \mathcal B^{(k+1)}$ is nonempty. Indeed, since $F \cap \mathcal B^{(k)}$ is an infinite compact set, it has a cluster point, and this point belongs to $\mathcal B^{(k+1)}$.

\begin{lemma}
\langlebel{system}
Let $n, k, l \in e^\astn$, $k \leq l$, $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is finite, and let $x_1, \ldots, x_n$ be distinct points in $\mathcal W_{\mathcal H_1}^{(l)}$.
Let $e_1, \ldots, e_n$ be nonzero vectors in $E_1$ with $\max_{j=1, \ldots , n} \norm{e_i}=1$, and $U_1, \ldots, U_n$ be pairwise disjoint open sets such that $x_j \in U_j$ for each $j=1, \ldots, n$. 
Suppose that for a given $0<e^\astp<1$, there exist functions $h_1, \ldots, h_n \in \mathcal A_1^{E_1}$ satisfying that for each $j=1, \ldots, n$, 
\[0 \leq h_j \leq 1, h_j(x_j)>1-e^\astp, h_j<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j,\] 
and such that the set $\mathcal R(u^\astum_{j=1}^n T(h_j \otimes e_j), e^\astp) \cap \mathcal B^{(k)}$ is empty. 

Then there exist functions $(h_1^i)_{i=0}^{l}, \ldots , (h_n^i)_{i=0}^{l}$ in $\mathcal A_1^{E_1}$ satisfying that for each $j=1, \ldots, n$ and $i=0, \ldots, l$,  
\[0 \leq h_j^i \leq 1, h_j^i<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j, h_j^i<h_j^{i+1}+e^\astp v^\astext{ for } i<l,\] 
$h_j^i(x_j^i) \geq 1-e^\astp$ for some $x_j^i \in \mathcal W_{\mathcal H_1}^{(i)}$ and such that for every $y \in \mathcal B$, the set \[\{ i \in \{0, \ldots , l\}: y \in \mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp)\}\]
 has cardinality at most $k$.
e^\astnd{lemma}

\begin{proof}
The Lemma will be proved once we verify that the following Claim is true. 
		
e^\astmph{Claim.
For each $i=l, l-1, \ldots, 0$ there exist functions
\[\{h_1^l, \ldots, h_n^l\}, \{h_1^{l-1}, \ldots, h_n^{l-1}\} \ldots , \{h_1^i, \ldots, h_n^i\}\] in $\mathcal A_1^{E_1}$ and points 
\[\{x_1^l, \ldots, x_n^l\}, \{x_1^{l-1}, \ldots, x_n^{l-1}\}, \ldots, \{x_1^i, \ldots, x_n^i\}\] such that for each $j=1, \ldots, n$ and $m=i, \ldots, l$, $x_j^m \in \mathcal W_{\mathcal H_1}^{(m)}$,
\[0 \leq h_j^m \leq 1, h_j^m<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j,
h_j^m(x_j^m) \geq 1-e^\astp, h_j^m<h_j^{m+1}+e^\astp v^\astext{ for } m<l\] and such that
\begin{equation}
\langlebel{empty}
 \bigcup_{s=0}^{l-i} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p   \otimes e_j), e^\astp)\}=e^\astmptyset. 
e^\astnd{equation}}

Indeed, assume first that the Claim holds. This in particular for $i=0$ and $s=k$ means that 
\[e^\astmptyset=\mathcal B \cap \bigcup_{0 \leq r_1< \ldots <r_{k+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{k+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p   \otimes e_j), e^\astp).\]
This means that for each $0 \leq r_1 < \ldots <r_{k+1} \leq l$, the intersection 
\[\mathcal B \cap \bigcap_{p \in \{r_1, \ldots , r_{k+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\] is empty, which we wanted to prove. 

e^\astmph{Proof of Claim.}
We proceed by reverse induction for $i=l, l-1, \ldots, 0$. For the case $i=l$, it is enough to denote $h_j^l=h_j$ and $x_j^l=x_j$ for $j=1, \ldots, n$, since then e^\astqref{empty} holds due to the assumption that
$\mathcal R(u^\astum_{j=1}^n T(h_j \otimes e_j), e^\astp) \cap \mathcal B^{(k)}$ is empty.

 Thus suppose that $0 \leq i<l$, and that we have found the functions \[\{h_1^l, \ldots, h_n^l\}, \{h_1^{l-1}, \ldots, h_n^{l-1}\} \ldots , \{h_1^{i+1}, \ldots, h_n^{i+1}\}\] in $\mathcal A_1^{E_1}$ and points 
 \[\{x_1^l, \ldots, x_n^l\}, \{x_1^{l-1}, \ldots, x_n^{l-1}\}, \ldots, \{x_1^{i+1}, \ldots, x_n^{i+1}\}\] 
 such that e^\astqref{empty} holds for $i+1$, and satisfying all the other above conditions. 
Hence we know that the set 

\[ \bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}\]
is empty. Since for each $s=0, \ldots, l-i-1$, the set \[\bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\] is compact, the set
\[ \mathcal B^{(\max\{k-s-1, 0\})} \cap \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\]
is finite (see the note above the statement of the lemma). 

The set $\mathcal B^{(k)}  \cap \Omega_{\mathcal H_2}$ is also finite by the assumption, hence so is the set
\begin{equation}
\nonumber
\begin{aligned}
G=&(\mathcal B^{(k)}  \cap \Omega_{\mathcal H_2})
\cup \\& 
\bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s-1, 0\})} \cap \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}.
e^\astnd{aligned}
e^\astnd{equation} 

Next, for each $j=1, \ldots n$, by the continuity of the function $h_j^{i+1}$ we find an open set $V u^\astubseteq U_j$ containing $x_j^{i+1}$, and such that 
$h_j^{i+1}>1-e^\astp$ on $V$. Next, since $x_j^{i+1} \in \mathcal W_{\mathcal H_1}^{(i+1)}$, we may find a sequence of points $\{z_j^m\}_{m \in e^\astn} u^\astubseteq \mathcal W_{\mathcal H_1}^{(i)}$ distinct from $x_j^{i+1}$, together with their pairwise disjoint open neighborhoods $\{U_j^{m}\}_{m \in e^\astn}$, all contained in $V$. Since each point $z_j^m$ is a weak peak point, we may find a function $g_j^m \in \mathcal A_1^{E_1}$ satisfying that
\[ 0 \leq g_j^m \leq 1, g_j^m(z_j^m)>1-Fr\'echet\ ac{e^\astp}{2^m} v^\astext{ and } g_j^m<Fr\'echet\ ac{e^\astp}{2^m} v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j^m.\]
Then for each $m \in e^\astn$, $g_j^m \leq h_j^{i+1}+e^\astp$ by Lemma |ef{nerovnost}. Using Lemma |ef{finite}, for each $j=1, \ldots, n$ we find an index $m_j$ such that the function $g_j^{m_j}$ satisfies that $\mathcal R(T(g_j^{m_j} \otimes e_j, Fr\'echet\ ac{e^\astp}{n}) \cap G$ is empty, and we denote $h_j^i=g_j^{m_j}$ and $x_j^i=z_j^{m_j}$. Thus for $y \in G$ it holds that
\[\norm{u^\astum_{j=1}^n T(h_j^i \otimes e_j)(y)} \leq u^\astum_{j=1}^n \norm{T(h_j^i \otimes e_j)(y)} < nFr\'echet\ ac{e^\astp}{n}=e^\astp,\]
and hence $\mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp) \cap G$ is empty. Consequently, 
\begin{equation}
\nonumber
\begin{aligned}
e^\astmptyset&=\mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp) \cap G=
\mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp) \cap  \mathcal Big((\mathcal B^{(k)}  \cap \Omega_{\mathcal H_2}) \cup \\& \bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s-1, 0\})} 
\cap \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}\mathcal Big)
=\\&=
(\mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp) \cap \mathcal B^{(k)}) \cup  \bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s-1, 0\})} 
\cap \\& \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp) \cap \mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp)\}
=\\&=
(\mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), e^\astp) \cap \mathcal B^{(k)}) \cup \\&  \bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s-1, 0\})} 
\cap \bigcup_{i=r_1< \ldots <r_{s+2} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+2}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}
=\\&=
\bigcup_{s=-1}^{l-i-1} \{\mathcal B^{(\max\{k-s-1, 0\})} 
\cap \bigcup_{i=r_1< \ldots <r_{s+2} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+2}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}
=\\&=
\bigcup_{s=0}^{l-i} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i=r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}.
e^\astnd{aligned}
e^\astnd{equation}
Thus recalling the inductive assumption we conclude that the set
\begin{equation}
\nonumber
\begin{aligned}
&\bigcup_{s=0}^{l-i} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}=\\&
\bigcup_{s=0}^{l-i} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i=r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}
\cup \\& \cup \bigcup_{s=0}^{l-i-1} \{\mathcal B^{(\max\{k-s, 0\})} \cap \bigcup_{i+1 \leq r_1< \ldots <r_{s+1} \leq l} \bigcap_{p \in \{r_1, \ldots , r_{s+1}\}} \mathcal R(u^\astum_{j=1}^n T(h_j^p \otimes e_j), e^\astp)\}
e^\astnd{aligned}
e^\astnd{equation}
is empty. This finishes the induction step and the proof. 
e^\astnd{proof}

Next, similarly as in section 3 we define certain sets $\mathcal G$, $\mathcal Theta$, which we use for the proof of Theorem |ef{distance}. For $s, m \in e^\astn$, $m < s$, $e^\astp>0$ and functions $h^1, \ldots , h^s \in \mathcal A_1^{E_1}$ we denote
\begin{equation}
\nonumber
\begin{aligned}
 &\mathcal{G}(h^1, \ldots, h^s, m, e^\astp)=\{ (g^1, \ldots, g^m) \in (\mathcal A_1^{E_1})^m, \\& \forall i=1, \ldots, m: 0 \leq g^i \leq 1, g^i \geq h^i-e^\astp v^\astext{ and } g^i \leq h^{s-m+i}+e^\astp\}. 
e^\astnd{aligned}
e^\astnd{equation}

Next, for $m, n, s \in e^\astn$, $m < s$, $e^\astp>0$, vectors $e_1, \ldots, e_n \in E_1$ and functions \\$(h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s$ from $\mathcal A_1^{E_1}$ we denote
\begin{equation}
\nonumber
\begin{aligned}
& \mathcal Theta(T, (h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s, (e_j)_{j=1}^n, m, e^\astp) = \\& \bigcap \{ \mathcal R(T(u^\astum_{j=1}^n u^\astum_{i=1}^m g_{j}^i \otimes e_j), e^\astp): \forall j=1, \ldots n, (g_j^1, \ldots, g_j^m) \in \mathcal{G}(h_j^1, \ldots, h_j^s, m, e^\astp) \}.
e^\astnd{aligned}
e^\astnd{equation}

Suppose that the functions $(h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s$ from $\mathcal A_1^{E_1}$ satisfy that for each $j=1, \ldots, n$ and $r=1, \ldots, s$, $0 \leq h_j^r \leq 1$ and for $r<s$, $h_j^r<h_j^{r+1}+e^\astp$. Then for $v^\astilde{e^\astp} \geq (s-1)e^\astp$ it holds that $h_j^r \leq h_j^t+v^\astilde{e^\astp}$ for each $r, t \in \{1, \ldots, s\}, r<t$. Consequently, for each $j=1, \ldots, n$ and $m \leq s$, the set $\mathcal{G}(h_j^1, \ldots, h_j^s,m, v^\astilde{e^\astp})$ is nonempty. Indeed, for each $m$ indices $r_1, \ldots, r_m \in \{1, \ldots, s\}$, 
\begin{equation}
\langlebel{nalezi}
(h_j^{r_1}, \ldots, h_j^{r_m}) \in \mathcal{G}(h_j^1, \ldots, h_j^s,m, v^\astilde{e^\astp}).
e^\astnd{equation} 
Thus the set of functions defining 
\[\mathcal Theta(T, (h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s, (e_j)_{j=1}^n, m, v^\astilde{e^\astp})\] is also nonempty. 

The following Lemma, which is an another crucial ingredient for the proof of Theorem |ef{distance}, is inspired by \cite[Lemma 2.1]{CANDIDOc0}.

\begin{lemma}
\langlebel{nonempty}
Let $n, m, s \in e^\astn$, $m<s$, $\norm{T^{-1}}=1$, $\norm{T}<Fr\'echet\ ac{s+m}{s-m}$. Choose $e^\astp \in (0, Fr\'echet\ ac{1}{n})$ satisfying
\[e^\astp<Fr\'echet\ ac{s+m-\norm{T}(s-m)}{\norm{T}(2m+n(s+5m))+n(s+m)+2}\]
and let nonzero vectors $e_1, \ldots, e_n \in E_1$ satisfy that $\max_{i=1, \ldots , n} \norm{e_i}=1$. Let $U_1, \ldots, U_n$ be pairwise disjoint open sets in $K_1$, and functions $(h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s$ from $\mathcal A_1^{E_1}$ satisfy for each $r=1, \ldots, s$ and $j=1, \ldots, n$,
\[0 \leq h_j^r \leq 1, h_j^r<e^\astp v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j,  
v^\astext{ and for } r<s, h_j^r \leq h_j^{r+1} +Fr\'echet\ ac{e^\astp}{s}.\]
Choose $j_0 \in \{1, \ldots, n\}$ satisfying that $\norm{e_{j_0}}=1$. If $\norm{h_{j_0}^1} > 1-Fr\'echet\ ac{e^\astp}{s}$, then 
\[\mathcal Theta(T, (h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s, (e_j)_{j=1}^n, m, e^\astp) \cap \mathcal B\] is nonempty.
e^\astnd{lemma}

\begin{proof}
By the assumptions, there exists a point $x_0 \in \mathcal Ch_{\mathcal H_1} K_1$ satisfying that
\[h_{j_0}^r(x_0) \geq 1-(s-1)Fr\'echet\ ac{e^\astp}{s}>1-e^\astp, \quad r=1, \ldots, s. \]
For $j \neq j_0$ and $r=1, \ldots, s$ we have $h_{j}^r(x_0)<e^\astp$ by disjointness of the sets $U_1, \ldots , U_n$. Consequently, since $\norm{T^{-1}}=1$, we have
\begin{equation}
\nonumber
\begin{aligned}
&\norm{T \mathcal Big(u^\astum_{j=1}^n(2u^\astum_{r=1}^m h_{j}^r+u^\astum_{r=m+1}^s h_j^r) \otimes e_j\mathcal Big)} \geq
\norm{u^\astum_{j=1}^n(2u^\astum_{r=1}^m h_{j}^r+u^\astum_{r=m+1}^s h_j^r) \otimes e_j)} \geq \\& \geq 
\norm{(u^\astum_{j=1}^n(2u^\astum_{r=1}^m h_{j}^r+u^\astum_{r=m+1}^s h_j^r) \otimes e_j)(x_0)} \geq
\norm{((2u^\astum_{r=1}^m h_{j_0}^r+u^\astum_{r=m+1}^s h_{j_0}^r) \otimes e_{j_0})(x_0)}-\\&\norm{(u^\astum_{j \neq j_0}(2u^\astum_{r=1}^m h_{j}^r+u^\astum_{r=m+1}^s h_j^r) \otimes e_j)(x_0)}>\\&> (2m+s-m)(1-e^\astp)-(n-1)(2m+s-m)e^\astp=(s+m)(1-ne^\astp).
e^\astnd{aligned}
e^\astnd{equation} 

Thus there exists a point $y \in \mathcal B$ such that
\[ \norm{T(u^\astum_{j=1}^n(2u^\astum_{r=1}^m h_{j}^r+u^\astum_{r=m+1}^s h_j^r) \otimes e_j)(y)} > (s+m)(1-ne^\astp).\]

We claim that $y \in \mathcal Theta(T, (h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s, (e_j)_{j=1}^n, m, e^\astp)$. Thus we choose arbitrary functions $(g_1^i)_{i=1}^m, \ldots, (g_n^i)_{i=1}^m$ in $\mathcal A_1^{E_1}$ satisfying that for each $j=1, \ldots n$, $(g_j^1, \ldots, g_j^m) \in \mathcal{G}(h_j^1, \ldots, h_j^s, m, e^\astp)$ (we know from e^\astqref{nalezi} that set of such functions is nonempty). Fix $j \in \{1, \ldots, n\}$. By definition of the set $\mathcal{G}(h_j^1, \ldots, h_j^s, m, e^\astp)$ we have 
\begin{equation}
\langlebel{prvni}
(u^\astum_{i=1}^m h_j^i)-me^\astp \leq u^\astum_{i=1}^m g_j^i
e^\astnd{equation}
and
\begin{equation}
\langlebel{druha}
u^\astum_{i=1}^m g_j^i \leq (u^\astum_{i=s-m+1}^s h_j^i)+me^\astp.
e^\astnd{equation}
Consequently,
\begin{equation}
\nonumber
\begin{aligned}
-(s-m)-2me^\astp&\leq -(u^\astum_{i=m+1}^s h_j^i)-2me^\astp 
\leq \\& \leq  (2u^\astum_{i=1}^{s-m} h_j^i-u^\astum_{i=m+1}^s h_j^i)-2me^\astp \leq^{e^\astqref{druha}} \\& \leq
2u^\astum_{i=1}^{s-m} h_j^i+2u^\astum_{i=s-m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i-u^\astum_{i=m+1}^s h_j^i=\\&
=2u^\astum_{i=1}^s h_j^i-2u^\astum_{i=1}^m g_j^i-u^\astum_{i=m+1}^s h_j^i= \\&=
2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i
\leq^{e^\astqref{prvni}} \\& \leq (u^\astum_{i=m+1}^{s} h_j^i)+2me^\astp \leq s-m+2me^\astp.
e^\astnd{aligned}
e^\astnd{equation}	
Thus \[\norm{(2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i) \otimes e_j} \leq s-m+2me^\astp.\]
Next, notice that for $j \in \{1, \ldots, n\}$, $i \in \{1, \ldots, m\}$ and $x \in \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j$ we have
$g_j^i(x) \leq h_j^{s-m+i}(x) +e^\astp < 2e^\astp$, hence \[\norm{((2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i) \otimes e_j)(x)}<2me^\astp+(s-m)e^\astp+4me^\astp=(s+5m)e^\astp.\]

Thus by Lemma |ef{omezenost} we have 
\begin{equation}
\nonumber
\begin{aligned}
&\norm{u^\astum_{j=1}^n(2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i) \otimes e_j}
\leq s-m+2me^\astp+n(s+5m)e^\astp.
e^\astnd{aligned}
e^\astnd{equation}

Now we claim that $\norm{T(u^\astum_{j=1}^n u^\astum_{i=1}^m g_{j}^i \otimes e_j)(y)} \geq e^\astp$. Indeed, otherwise we would have
\begin{equation}
\nonumber
\begin{aligned}
&\norm{T}(s-m+2me^\astp+n(s+5m)e^\astp)
\geq \\& \geq
\norm{T(u^\astum_{j=1}^n(2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i) \otimes e_j)}
\geq \\& \geq
\norm{T(u^\astum_{j=1}^n(2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i-2u^\astum_{i=1}^m g_j^i) \otimes e_j)(y)}
\geq \\& \geq 
\norm{T(u^\astum_{j=1}^n(2u^\astum_{i=1}^m h_j^i+u^\astum_{i=m+1}^s h_j^i) \otimes e_j)(y)}- 2\norm{T(u^\astum_{j=1}^nu^\astum_{i=1}^m g_j^i \otimes e_j))(y)}
 >\\&>
(s+m)(1-ne^\astp)-2e^\astp,
e^\astnd{aligned}
e^\astnd{equation}
contradicting the choice of $e^\astpsilon$. Thus $y \in \mathcal Theta(T, (h_1^r)_{r=1}^s, \ldots , (h_n^r)_{r=1}^s, (e_j)_{j=1}^n, m, e^\astp)$, and we are done.
e^\astnd{proof}

Now we are ready to prove Theorem |ef{distance}.

e^\astmph{Proof of Theorem |ef{distance}.}

\begin{proof}
We only need to prove that $\norm{T}\norm{T^{-1}} \geq Fr\'echet\ ac{2l+2-k}{k}$, since the bound $3$ follows from statements a), b) and c) of Theorem |ef{jednostranna}. Let $\mathcal A_1$ be the canonical scalar function space associated to $\mathcal H_1$.

e^\astmph{Claim. There exists $n \in e^\astn$ such that for each $\zeta>0$, there exist distinct points $(x_j)_{j=1}^n \in \mathcal W_{\mathcal H_1}^{(l)}$,
vectors $e_1, \ldots, e_n$ in $E_1$ with $\max_{j=1, \ldots , n} \norm{e_i}=1$, $U_1, \ldots, U_n$ pairwise disjoint open sets such that $x_j \in U_j$ for each $j=1, \ldots, n$, and functions $h_1, \ldots, h_n \in \mathcal A_1^{E_1}$ satisfying that for each $j=1, \ldots, n$, 
\[0 \leq h_j \leq 1, h_j(x_j)>1-\zeta, h_j<\zeta v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j\] 
and such that the set $\mathcal R(u^\astum_{j=1}^n T(h_j \otimes e_j), \zeta) \cap \mathcal B^{(k)}$ is empty.}

Suppose first that the Claim holds, and assume for a contradiction that the isomorphism $T$ satisfies that $\norm{T^{-1}}=1$ and $\norm{T}<Fr\'echet\ ac{2l+2-k}{k}$. We find an $n \in e^\astn$ from the Claim, and we choose $e^\astp \in (0, Fr\'echet\ ac{1}{n})$ satisfying

\[e^\astp<Fr\'echet\ ac{2l+2-k-\norm{T}k}{\norm{T}(2l+2-2k+n(6l+6-5k))+n(2l+2-k)+2} v^\astext{, and let } \zeta = Fr\'echet\ ac{e^\astp}{l+1}.\]

Then we use the Claim to find all the above objects for this $\zeta$. Next, since in all cases (i), (ii) and (iii), the set $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is finite, we can use Lemma |ef{system} to find functions $(h_1^r)_{r=0}^{l}, \ldots , (h_n^r)_{r=0}^{l}\in \mathcal A_1^{E_1}$ satisfying that for each $j=1, \ldots, n$ and $i=0, \ldots, l$,  
\[0 \leq h_j^i \leq 1, h_j^i<\zeta v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j^i, h_j^i<h_j^{i+1}+\zeta v^\astext{ for } i<l,\] 
$h_j^i(x_j^i) \geq 1-\zeta$ for some $x_j^i \in \mathcal W_{\mathcal H_1}^{(i)}$ and such that for every $y \in \mathcal B$,
\begin{equation}
\langlebel{maxk}
\abs{\{ i \in \{0, \ldots , l\}: y \in \mathcal R(u^\astum_{j=1}^n T(h_j^i \otimes e_j), \zeta)\}} \leq k.
e^\astnd{equation}
Next, since $\zeta=Fr\'echet\ ac{e^\astp}{l+1}$, we can use Lemma |ef{nonempty} for $s=l+1$ and $m=l-k+1$ to obtain a point 
\[y \in \mathcal Theta(T, (h_1^r)_{r=0}^{l}, \ldots , (h_n^r)_{r=0}^{l}, (e_j)_{j=1}^n, l-k+1, e^\astp) \cap \mathcal B.\]
 By e^\astqref{maxk}, there are at most $k$ indices $i$ from the set $\{0, \ldots , l\}$ satisfying that $\norm{u^\astum_{j=1}^n T(h_j^i \otimes e_j)(y)} \geq \zeta$. Thus we can fix $i_1, \ldots, i_{l-k+1} \in \{0, \ldots, l\}$ such that for each $p=1, \ldots , l-k+1$ it holds that $\norm{u^\astum_{j=1}^n T(h_j^{i_p} \otimes e_j)(y)}<\zeta$. But then, since $\zeta=Fr\'echet\ ac{e^\astp}{l+1}$, 
 \[(h_j^{i_1}, \ldots, h_j^{i_{l-k+1}}) \in \mathcal{G}(h_j^0, \ldots, h_j^l, l-k+1, e^\astp)\]
 for each $j=1, \ldots, n$, see e^\astqref{nalezi}. 
 
 Thus by the definition of $\mathcal Theta(T, (h_1^r)_{r=0}^{l}, \ldots , (h_n^r)_{r=0}^{l}, (e_j)_{j=1}^n, m, e^\astp)$ we know that \[\norm{T(u^\astum_{j=1}^nu^\astum_{p=1}^{l-k+1} h_j^{i_p} \otimes e_j)(y)} \geq e^\astp.\]
  Consequently, we have
\[e^\astp \leq \norm{T(u^\astum_{j=1}^nu^\astum_{p=1}^{l-k+1} h_j^{i_p} \otimes e_j)(y)} \leq u^\astum_{p=1}^{l-k+1} \norm{T(u^\astum_{j=1}^n h_j^{i_p} \otimes e_j)(y)} < (l-k+1)\zeta<e^\astp.\]  
This contradiction shows that $\norm{T}\norm{T^{-1}} \geq Fr\'echet\ ac{2l+2-k}{k}$. Thus to finish the proof it is enough to prove the above Claim.

e^\astmph{Proof of Claim.}\\
In case (i), we put $n=1$. Since $\mathcal W_{\mathcal H_1}^{(l)}$ is nonempty, we can take an arbitrary point $x_1 \in \mathcal W_{\mathcal H_1}^{(l)}$ and vector $e_1 \in S_{E_1}$, and then, since the set $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is empty, the function $h_1$ exists simply by the definition of a weak peak point. 

For the case (ii), we again put $n=1$, and we pick an arbitrary $e_1 \in S_{E_1}$. Since $\mathcal W_{\mathcal H_1}^{(l)}$ is infinite, there exist pairwise disjoint open sets $\{U_m \}_{m \in e^\astn}$ and points $\{ z_m\}_{m \in e^\astn} u^\astubseteq \mathcal W_{\mathcal H_1}^{(l)}$ such that $z_m \in U_m$ for each $m \in e^\astn$. For a given $\zeta>0$ we find functions $\{g_m\}_{m \in e^\astn} u^\astubseteq \mathcal A_1^{E_1}$ such that for each $m \in e^\astn$, 
\[0 \leq g_m \leq 1, g_m(z_m)>1-Fr\'echet\ ac{\zeta}{2^m} v^\astext{ and } g_m <Fr\'echet\ ac{\zeta}{2^m} v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_m.\] Now, since $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}$ is finite,  it is enough to use Lemma |ef{finite} to obtain the function $h_1=g_m$ and the point $x_1=z_m$ for a suitable $m \in e^\astn$.

Finally, for the case (iii) we suppose that $\mathcal W_{\mathcal H_1}^{(l)}=\{ x_1, \ldots, x_n\}$ and $\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}=\{y_1, \ldots, y_m\}$. Given $\zeta>0$, we find pairwise disjoint open sets $U_1, \ldots, U_n$ such that $x_j \in U_j$ for each $j=1, \ldots, n$, and functions $h_1, \ldots, h_n$ in $\mathcal A_1^{E_1}$ satisfying that 
\[0 \leq h_j \leq 1, h_j(x_j)>1-\zeta v^\astext{ and } h_j<\zeta v^\astext{ on } \mathcal Ch_{\mathcal H_1} K_1 u^\astetminus U_j, \quad j=1, \ldots, n.\]

As in the proof of Theorem |ef{jednostranna}c), we define a linear operator $S:(E_1^n, \norm{\cdot}_{\max}) |ightarrow \mathcal C(\mathcal B^{(k)} \cap \Omega_{\mathcal H_2}, E_2) u^\astimeq (E_2^m, \norm{\cdot}_{\max})$ as
\[S(e_1, \ldots, e_n)=T(u^\astum_{i=1}^n h_i \otimes e_i)|_{(\mathcal B^{(k)} \cap \Omega_{\mathcal H_2})}.\] 
Then as in the proof of Theorem |ef{jednostranna}c), $S$ is bounded. Since by our assumption, $E_2^m$ does not contain an isomorphic copy of $E_1^n$, it follows that $S^{-1}$ is not bounded. Consequently, there exist nonzero vectors $e_1, \ldots, e_n \in E_1$ with $\max_{i=1, \ldots , n} \norm{e_i}=1$, such that $\norm{T(u^\astum_{i=1}^n h_i \otimes e_i)|_{(\mathcal B^{(k)} \cap \Omega_{\mathcal H_2})}} <\zeta$, and those are the vectors we were looking for. The proof is finished.

e^\astnd{proof}

e^\astnd{document}