Néron-Severi lattice by considering the complement of $M$, namely
$\text{NS}(X)=M\oplus M^{\perp}$. Thus, we could use the tool of lattice
polarizations, and take $L$ to be (isometric to) a direct sum of lattices of
ADE type. In this way, $L$ would play the role of $M$ in the decomposition of
$\text{NS}(X)$. Hence, the lattice polarization would specify the intersection
structure of the exceptional divisors, and in turn would characterize the ADE
type of the singularities on the $K3$ surface. This whole procedure allows us
to complement the cobordism conjecture with the information that it does not
provide. We will come back to this particular case in the next section.

Now, we need a way to find out which ADE lattices can polarize a $K3$ surface,
and if, further, they can be realized as coming from the singularities of an
elliptically fibered $K3$. This in fact not an easy question, since we have
the requirement that $i(L)$ must contain a pseudo-ample class (which we have
in fact not defined). One of the first conditions that we can anticipate for
our polarizing lattice is that, due to the requirement that its signature must
be that of a so-called hyperbolic lattice, namely $(1,t)$. Hence, purely
positive-definite or negative-definite lattices are off the table, as those
would have signatures $(n,0)$ and $(0,n)$, respectively. One way to include
them would be to consider their direct sum with $H$, the unique hyperbolic
rank 2 lattice which we introduced a few paragraphs above. Namely, we would
look for lattices $L$ which decomposed as $L=H\oplus M$, with $M$ some
positive or negative definite lattice. In fact, it turns out that a lattice
polarization is equivalent to an elliptic fibration whenever the polarizing
lattice $L$ admits a decomposition $L=H\oplus M$ (or is isometric to one such
decomposition).

One publication that might shed some light with this issue is [Nik83, Table 1,
pg. 1434]. In particular, the table that we have cited contains all of the
hyperbolic, even, 2-elementary lattices which admit a primitive embedding into
$\Lambda_{K3}$. A 2-elementary lattice $L$ is one in which its discriminant
group, $L^{*}/L$ satisfies

$L^{*}/L\simeq(\mathbb{Z}/2\mathbb{Z})^{a}$ (111)

Here, $L^{*}$ denotes the dual lattice, and we have used the fact that due to
the symmetric bilinear product there is an inclusion $L\subseteq L^{*}$. In
the publication we have cited, these kind of lattices are characterized by
three integers, namely $(\text{rk}(L),a(L),\delta_{L})$. The first is the rank
of the lattice, the second one is related to the discriminant group as
above141414Note that in the publication this parameter is denoted by
$l(A_{S})$, with $S$ being the lattice in question., and $\delta_{L}$, defined
as follows. For an even, 2-elementary lattice $L$, consider its dual, $L^{*}$,
and consider the pairing on $L^{*}$ defined by the extension of the pairing on
$L$ by means of the inclusion $L\subseteq L^{*}$. Then

$\delta_{L}=\begin{cases}0\ \text{if
}\langle\alpha,\alpha\rangle\in\mathbb{Z}\ \forall\ \alpha\in L^{*}\\\ 1\
\text{otherwise}\end{cases}$

With the above defined data, one is able to determine which lattices can and
cannot be embedded into $\Lambda_{K3}$. This is done for example in the
original reference [Nik83], but also in the more recent [CM21]. We will work
off of the latter reference from now on, as it contains a broader set of data.

###  9.4 Restrictions on the allowed ADE singularities from lattice
polarizations and the cobordism conjecture.

We will now try to put everything together. Firstly, as we have said, the
cobordism conjecture states that in order for a $K3$ surface to be a valid
background for a theory of quantum gravity, it must come accompanied by a
connected sum of 16 copies of $\mathbb{CP}^{2}$. Namely, the $K3$ surface that
we must consider is $\hat{X}=X\\#\mathbb{CP}^{2}\\#\stackrel{{\scriptstyle
16)}}{{\ldots}}\\#\mathbb{CP}^{2}$, with $X$ being our original $K3$ surface.
We used the fact that a connected sum of some manifold $X$ with copies of
$\mathbb{CP}^{2}$ is diffeomorphic to $X$ blown up a certain number of times
to interpret this as the condition that the original $K3$ surface must be
blown up exactly 16 times. From our discussion of line bundles and blow-ups,
we know that this produces 16 exceptional divisors, which span a lattice
$M=\text{Span}_{\mathbb{Z}}\\{E_{1},\ldots,E_{16}\\}$. The intersection
structure of these divisors then determines the ADE type of the singularities
that were blown-up. However, the cobordism conjecture contains no such
information about the intersection structure of the divisors. Hence, we
resorted to lattice polarizations in order to try to extract some conditions
on the allowed ADE type of the singularities.

As we have said before, from the perspective of lattice polarizations, the
presence of a factor of $H$, the unique hyperbolic rank 2 lattice, whose
intersection form is given by (107) is equivalent to the fact that the $K3$
surface is elliptically fibered. Moreover, as we have preiously stated, the
fact that the signature of this lattice is $(1,1)$ allows for polarizing
lattices which are not hyperbolic, but rather positive or negative definite,
by considering their direct sum with $H$. Thus, we may break the polarizing
lattice down as $L=H\oplus M$, where $M$ is the lattice which determines the
ADE structure of the singularities. Then, the cobordism conjecture states that
$\text{rk}(M)=16$, and therefore the polarizing lattice must have
$\text{rk}(L)=18$.

A priori, the fact that the rank of the ADE part of the polarizing lattice
should have rank 16 should start ringing some bells. Indeed, there are two
such lattices which are of great importance for string theory, as they are the
only two lattices which are allowed as gauge groups in heterotic string
theory. If we further require that the automorphism group of the resulting
(resolved) $K3$ surface be finite, we can check in [CM21, Table 3] that the
two possibilities for the polarizing lattice $L=H\oplus K$ are indeed

$K=\begin{cases}(-E_{8})\oplus(-E_{8}),\\\ -D_{16}.\end{cases}$ (112)

Note that the minus signs come form the fact that the curves over which the
polarized lattice acts must be of self-intersention $-2$. Given the structure
of $\Lambda_{K3}$, it was expected that, if the ranks matched, we could embed
two copies of $E_{8}$ into it. Lastly, we comment on one piece of data present
on the table we have referenced. In this publication, an explicit difference
is made between $K$, the polarizing lattice itself, and $K^{\text{root}}$, the
sub-lattice spanned by the roots of $K$. Namely, $K^{\text{root}}\subseteq K$
is the lattice spanned by the elements of self intersection $-2$. The quotient
between the two is denoted by151515We note in passing that this quotient is
related to the Mordell-Weil group of the elliptic fibration.
$\mathcal{W}\coloneqq K/K^{\text{root}}$. For the two cases above, we have
that for $(-E_{8})\oplus(-E_{8})$, $\mathcal{W}=1$, and hence both lattices
are one and the same. However, for $D_{16}$, $\mathcal{W}=\mathbb{Z}_{2}$.
Since $D_{16}$ is the lattice defined by the Lie algebra corresponding to
$SO(32)$, the root lattice is in fact given by $SO(32)/\mathbb{Z}_{2}$. Thus,
the two gauge groups associated to the singularities for a $K3$ surface under
the assumptions of the cobordism conjecture (assuming that the automorphism
group is finite) are precisely the only ones allowed for the heterotic string.

To close this chapter off, we note that the rank of the polarizing lattice
($\text{rk}(L)=18$) does not exhaust the upper bound on the rank of
$\text{NS}(X)$, which is $\rho(X)\leqslant 20$. Hence, there are still some
possibilities to ”fill out” the remainder of the polarizing lattice.

##  10 Other scenarios of interest

Staying in the $SO$-structure, we could also consider $\Omega_{8}^{SO}$, which
is generated by $C\mathbb{P}^{4}$ and $C\mathbb{P}^{2}\times C\mathbb{P}^{2}$.
This could give us more freedom to play with the two generators to kill the
cobordism class.

If we decide to leave the $SO$-structure, we could apply the same logic to
more complicated structures, such as $\Omega_{d}^{\text{Spin}}$,
$\Omega_{d}^{\text{String}}$, etc. As shown in [ACC22a], these particular ones
mentioned can be obtained by further constraining the vanishing of certain
charactersitic numbers. For example, $\Omega_{d}^{\text{Spin}}$ is obtained
from $\Omega^{SO}_{d}$ by requiring that $w_{2}=0$, as expected.

##  11 Conclusions and Outlook

In our work, we have applied the techniques of the Ricci flow equations with
surgery, as introduced by Hamilton and later brought to full fruition by
Perelman, and tried to combine them with the techniques of algebraic geometry
in order to study possible implications of the cobordism conjecture. After
having given a reasonable introduction to the relevant mathematical tools, it
was argued -and explicitly shown for the simple cases of Einstein manifolds
and Ricci solitons- that Ricci flow preserves Ponryagin and Stiefel-Whitney
numbers. This holds even in the presence of a ”neckpinch”, as the proper
treatment of it under the Ricci flow is to perform a particular type of
surgery: substituting the pinched neck by two smooth caps. Since these
characteristic numbers unambiguously fix an element of $\Omega_{4}^{SO}$, the
above statement implies that Ricci flow preserves the corresponding cobordism
class. It is nevertheless interesting and reasonable, from a more physical
perspective, to investigate the effects of completely removing one of the two
components resulting from the surgery. In general, this can imply the
necessity of balancing such a modification out by the introduction of defects,
that mimic the contribute of the removed component to the overall cobordism
class. Therefore, focusing on the specific string-theory inspired example of a
$K3$, it was argued that its cobordism class can be trivialised by adding to
it the connected sum of 16 copies of $\mathbb{CP}^{2}$.

In the later sections of the present work, the trivialization of the $K3$
class in $\Omega^{SO}_{4}$ is studied from two complementary perspectives. On
the one hand, we tried to apply the Ricci flow equations to the resulting
manifold, in order to see if we could recover the topological defects (as the
shrinking of parts of the manifold to points) by means of the flow. That is,
to see if the Ricci flow is the bridge connecting the perspectives of adding
connected sums of manifolds and adding topological defects. This was done by
means of a metric constructed on $\mathbb{CP}^{2}\\#\stackrel{{\scriptstyle
n)}}{{\ldots}}\\#\mathbb{CP}^{2}$, obtained by Claude Lebrun as a
generalization of the so-called Burns metric. We give arguments as to why this
manifold shrinks under the Ricci flow, thus producing singularities and
infinite distance limits in the moduli space of space-time metrics, of the
kind which are discussed in the context of the Swampland distance conjecture.

On the other hand, after the realization that taking connected sums of a
manifold with $\mathbb{CP}^{2}$ is equivalent to considering a blow-up of said
manifold, and that the singularities of $K3$ surfaces are classified by the
ADE Lie Algebras, we try to complement the information given by the cobordism
conjecture. Namely, we realize that we can recover the singularities by
flowing the $\mathbb{CP}^{2}$s that they produce, but we realize that the
cobordism conjecture contains no information whatsoever regarding the ADE
nature of these singularities. Hence, this is a piece of data that we must
introduce by hand, and might tell us about possible restrictions on the ADE
nature of the singularities. After introducing the necessary tools from
algebraic geometry, and reviewing how the ADE classification comes about, we
propose that this extra information might be added by means of lattice
polarizations of $K3$ surfaces. Upon doing so, and given the conditions
imposed by the cobordism conjecture, we recover the only two groups which are
allowed as gauge groups for the heterotic string.

While the technique of lattice polarizations is best understood for $K3$
surfaces, this work could be expanded by considering more complicated
structures, such as those on $\Omega_{d}^{\text{spin}}$ or
$\Omega_{d}^{string}$. In this context, one could find which manifolds are
needed to trivialize the bordism class, and try to apply the Ricci flow
equations to them. This would be easiest when staying in the $SO$-structure,
since we know that $\Omega_{8}^{SO}$ is generated by $\mathbb{CP}^{4}$ and
$\mathbb{CP}^{2}\times\mathbb{CP}^{2}$. Not only are these manifolds well-
known, but also the fact that we have more than one generator might allow us
more freedom to encounter different scenarios for the trivialization of a
given bordism class.

## Acknowledgments

The work of D.L. is supported by the Origins Excellence Cluster and by the
German-Israel-Project (DIP) on Holography and the Swampland.

##  12 Appendix: The (co)homology of $\mathbb{CP}^{2}\\#\mathbb{CP}^{2}$

In this somewhat more technical section, we will work out the homology and
cohomology groups of $\mathbb{CP}^{2}\\#\mathbb{CP}^{2}$. We will use a
collection of results which are fundamental in the context of algebraic
topology, but nonetheless require some degree of familiarity with the topic.
Due to this technical nature, we will not introduce these concepts here, but
rather point the reader to standard references such as [Hat00].

To begin the discussion, we note that the orientation of an $n$-dimensional
manifold does not affect its homological structure, but rather only the choice
of generator of the top homology group $H_{n}$. We refer to this generator as
the fundamental class of the manifold. Hence, in particular, we have that
$H_{p}(\mathbb{CP}^{2})=H_{p}(\overline{\mathbb{CP}}^{2})$, for
$p=1,\ldots,4$, all other homology groups being automaticall trivial for
dimensional reasons. Furthermore, as a consequence of the CW structure of
$\mathbb{CP}^{2}$, we have that

$H_{p}(\mathbb{CP}^{2})=\begin{cases}\mathbb{Z}\hskip 28.45274pt&\text{if
}p=0,2,4\\\ 0&\text{else}.\end{cases}$ (113)

However, in order to simplify calculations, we will not use the above
(singular) homology, but the so-called reduced homology, which we denote by
$\tilde{H}_{p}$. In this context of path connected spaces, this simply means
that we can set $\tilde{H}_{p}(\mathbb{CP}^{2})=0$, and hence

$\tilde{H}_{p}(\mathbb{CP}^{2})=\begin{cases}\mathbb{Z}\hskip
28.45274pt&\text{if }p=2,4\\\ 0&\text{else}.\end{cases}$ (114)

In the following, we will drop the tilde, as we will only be considering
reduced homology groups.

The actual computation of the homology groups of
$\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$ will be a two-step process fo
sorts. First, we will consider the long exact sequence of the pair
$(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})$. In said pair,
all of the points on the $\mathbb{S}^{3}$ over which we glue to define the
connected sum are identified as one, and thus
$(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\simeq\mathbb{CP}^{2}\vee\overline{\mathbb{CP}}^{2}$,
where $\vee$ denotes the one-point union of two spaces. Once this is done, we
will use a simple Mayer-Vietoris argument that will allow us to directly
compute the homology groups of our original space in terms of those of the
original $\mathbb{CP}^{2}$ factors.

The long exact sequence of the pair
$(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})$ is given by

$\ldots\longrightarrow
H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\stackrel{{\scriptstyle\partial_{*}}}{{\longrightarrow}}H_{p}(\mathbb{S}^{3})\longrightarrow
H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\longrightarrow
H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\longrightarrow\ldots$
(115)

Because the homology groups of spheres are

$H_{p}(\mathbb{S}^{n})=\begin{cases}\mathbb{Z}\hskip 28.45274pt&\text{if
}p=n\\\ 0&\text{else}.\end{cases}$ (116)

Then, for $p\neq 3,4$, we have that

$\ldots\longrightarrow 0\longrightarrow
H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\longrightarrow
0\longrightarrow\ldots$ (117)

Therefore, by exactness of the sequence, the map in the middle is an
isomorphism, so $H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\simeq
H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\simeq
H_{p}(\mathbb{CP}^{2}\vee\overline{\mathbb{CP}}^{2})$ for $p\neq 3,4$. For the
special case $p=3,4$, the interesting piece of the long exact sequence looks
like

$\begin{split}\ldots\longrightarrow 0&\longrightarrow
H_{4}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\longrightarrow
H_{4}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\longrightarrow\mathbb{Z}\longrightarrow\\\
&\longrightarrow
H_{3}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\longrightarrow
H_{3}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})\longrightarrow
0\longrightarrow\ldots\end{split}$ (118)

For the moment, we leave this as is.

Now that we have related the homology groups of
$\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$ to those of
$\mathbb{CP}^{2}\bigvee\overline{\mathbb{CP}}^{2}$, we can calculate the
latter by a simple Mayer-Vietoris argument. Let
$U,V\subseteq\mathbb{CP}^{2}\bigvee\overline{\mathbb{CP}}^{2}$ be two subsets
such that

$\mathbb{CP}^{2}\bigvee\overline{\mathbb{CP}}^{2}=\text{Int}(U)\cup\text{Int}(V),$
(119)

and that $U\cap V\neq\emptyset$. To ensure that both of these conditions are
satisfied, we pick $U$ and $V$ to be neighbourhoods of $\mathbb{CP}^{2}$ and
of $\overline{\mathbb{CP}}^{2}$ in
$\mathbb{CP}^{2}\vee\overline{\mathbb{CP}}^{2}$ respectively. Furthermore, we
take the neighbourhood small enough so that $U\cap V$ is contractible to a
point (said point being the one that defines the one-point union). This will
imply that the homology groups of $U\cap V$ will all be trivial.

In the above conditions we have the so-called Mayer-Vietoris long exact
sequence:

$\ldots\longrightarrow H_{p}(U\cap V)\longrightarrow H_{p}(U)\oplus
H_{p}(V)\longrightarrow H_{p}(U\cup V)\longrightarrow H_{p-1}(U\cap
V)\longrightarrow\ldots$ (120)

As already stated before, we have that $U\cap V$ deformation retracts to a
point, and hence all of the homology groups are trivial (even
$H_{0}(\text{pt})$, since we are using reduced homology). Furthermore, both
$U$ and $V$ deformation retract to $\mathbb{CP}^{2}$ and
$\overline{\mathbb{CP}}^{2}$ respectively, so in total we have that

$\ldots\longrightarrow H_{p}(\text{pt})\longrightarrow
H_{p}(\mathbb{CP}^{2})\oplus H_{p}(\overline{\mathbb{CP}}^{2})\longrightarrow
H_{p}(\mathbb{CP}^{2}\vee\overline{\mathbb{CP}}^{2})\longrightarrow
H_{p-1}(\text{pt})\longrightarrow\ldots$ (121)

However, since all of the (reduced) homology groups of a point vanish, by
exactness of the sequence this implies that

$H_{p}(\mathbb{CP}^{2})\oplus H_{p}(\overline{\mathbb{CP}}^{2})\simeq
H_{p}(\mathbb{CP}^{2}\vee\overline{\mathbb{CP}}^{2})$ (122)

for all $p\in\mathbb{Z}$.

By the above argument, we can just plug $H_{p}(\mathbb{CP}^{2})\oplus
H_{p}(\overline{\mathbb{CP}}^{2})$ into (115). By previous arguments, we have
that for $p=0,1,2$ we have that
$H_{p}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\simeq
H_{p}(\mathbb{CP}^{2})\oplus H_{p}(\overline{\mathbb{CP}}^{2})$. However, the
really interesting case is that of (118). To study it, we will use the
following two facts [Hat00, Th. 3.26]:

  * •

If $\mathcal{M}$ is an orientable, $n$-dimensional space, then

$H_{n}(\mathcal{M})=\mathbb{Z}.$ (123)

The choice of generator is called the fundamental class of $\mathcal{M}$, and
determines its orientation.

  * •

The connected sum of orientable spaces is again orientable.

Combining these two results, we have that (118) can be written as

$0\longrightarrow\mathbb{Z}\longrightarrow\mathbb{Z}\oplus\mathbb{Z}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}H_{3}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\longrightarrow
0$ (124)

The key to the above sequence is precisely the map $\varphi$. If we can show
that it is surjective, then it follows by exactness that
$\text{Im}(\varphi)=\text{Ker}(\psi)$, and hence $\psi$ is the zero map.
Furthermore, and also by exactness, $\psi$ is itself surjective, for the last
map is by force the zero map. Thus, from the surjectivity of $\varphi$ it
would follow that

$H_{3}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})=0$ (125)

This would be the last piece in our calculation of the homology groups of
$\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$. The only difference with (36)
would be that there we used singular homology, so
$H_{0}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})$ is nontrivial.

The fact that $\varphi$ is surjective follows from the particular way in which
one constructs the long exact sequence of a pair. Given $A\subseteq X$,
$H_{p}(X,A)$ is defined as the $p$th homology group with respect to the
quotient space $X/A$, in which $A$ is identified to a point. Thus, cycles in
the pair $(X,A)$ either have no boundary, or their boundary lies in $A$. To
construct the sequence, the map $\partial_{*}:H_{p}(X,A)\to H_{p-1}(A)$ is
then defined by precisely taking the boundary in $A$.

Now consider the fundamental class of the pair
$(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2},\mathbb{S}^{3})$. It is that of
$\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$, but with a $\mathbb{S}^{3}$
identified. Therefore, in the total space, this fundamental class has a
boundary, which is precisely the fundamental class of $\mathbb{S}^{3}$. In
other words, one of the generators of the $\mathbb{Z}\oplus\mathbb{Z}$
(depending on which $\mathbb{CP}^{2}$ we take as a reference) gets mapped to
the generator of $H_{3}(\mathbb{S}^{3})$, thus making $\varphi$ surjective.

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# Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach
spaces

Anton Tselishchev  Department of Mathematics, Bar-Ilan University, Ramat-Gan
5290002, Israel  St. Petersburg Department, Steklov Math. Institute, Fontanka
27, St. Petersburg 191023 Russia<EMAIL_ADDRESS>
###### Abstract.

For any sequence of positive numbers $(\varepsilon_{n})_{n=1}^{\infty}$ such
that $\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$ we provide an explicit simple
construction of $(1+\varepsilon_{n})$-bounded Markushevich basis in a
separable Hilbert space which is not strong, or, in other terminology, is not
hereditary complete; this condition on the sequence
$(\varepsilon_{n})_{n=1}^{\infty}$ is sharp. Using a finite-dimensional
version of this construction, Dvoretzky’s theorem and a construction of
Vershynin, we conclude that in any Banach space for any sequence of positive
numbers $(\varepsilon_{n})_{n=1}^{\infty}$ such that
$\sum_{n=1}^{\infty}\varepsilon_{n}^{2}=\infty$ there exists a
$(1+\varepsilon_{n})$-bounded Markushevich basis which is not a Schauder basis
after any permutation of its elements.

###### Key words and phrases:

Schauder basis; Markushevich basis; Auerbach basis

###### 2020 Mathematics Subject Classification:

46B15, 46B04, 46C15

Research supported by ISF Grant No. 1044/21 and by Russian Science Foundation
grant no. 23-11-00171

##  1\. Introduction

###  1.1. The main definitions

The word “basis” has many different meanings in the Banach space theory. The
questions of existence of bases in Banach spaces which possess some nice
properties lie in the foundation of the geometry of Banach spaces, as well as
other fields of Analysis. We begin with presenting certain definitions that
will be used throughout the paper.

Let $X$ be a separable Banach space. A system of vectors
$\\{x_{n}\\}_{n=1}^{\infty}$ in $X$ is _complete_ in $X$ if
$\overline{\operatorname{span}}\\{x_{n}\\}_{n=1}^{\infty}=X$. It is called
_minimal_ if $x_{m}\not\in\overline{\operatorname{span}}\\{x_{n}\\}_{n\neq m}$
for every $m\geq 1$.

If $\\{x_{n}\\}_{n=1}^{\infty}$ is a complete and minimal system, then there
exists a unique sequence of _biorthogonal functionals_
$\\{x_{n}^{*}\\}_{n=1}^{\infty}$ in $X^{*}$, i.e.,
$x_{n}^{*}(x_{m})=\delta_{nm}$. In this case with any element $x\in X$ one can
associate a formal series

(1.1) $x\sim\sum_{n=1}^{\infty}x_{n}^{*}(x)x_{n}.$

If for every $x\in X$ this series converges to $x$, then
$\\{x_{n}\\}_{n=1}^{\infty}$ is called a _Schauder basis_. In this case there
exists a constant $C$ such that for any $N>0$

(1.2) $\Big{\|}\sum_{n=1}^{N}x_{n}^{*}(x)x_{n}\Big{\|}\leq C\|x\|.$

The minimal constant such that this inequality holds is called the _basis
constant_ of $(x_{n})_{n=1}^{\infty}$.

It was a long-standing open problem whether every separable Banach space has a
Schauder basis. It was solved in the negative by P. Enflo: in [Enf73] he
constructed the first example of a space without a Schauder basis.

It is natural therefore to consider some weaker notions of bases. It leads us
to the next definition: a complete and minimal system
$\\{x_{n}\\}_{n=1}^{\infty}$ in $X$ is called a _Markushevich basis_ , or _M-
basis_ , if the sequence of its biorthogonal functionals $\\{x_{n}^{*}\\}$ is
total, i.e., if $x_{n}^{*}(x)=0$ for every $n\geq 1$, then $x=0$. The word
“basis” here might be somewhat misleading: it does not mean that the series
(1.1) converges to $x$. Already in 1943 Markushevich in the paper [Mar43]
proved that every separable Banach space contains an M-basis.

An M-basis $\\{x_{n}\\}_{n=1}^{\infty}$ is called _bounded_ if there exists a
constant $C$ such that $\|x_{n}\|\cdot\|x_{n}^{*}\|\leq C$ for every $n\geq
1$. Since the inequality (1.2) for a Schauder basis implies
$\|x_{n}\|\cdot\|x_{n}^{*}\|\leq 2C$, every Schauder basis is a bounded
M-basis. Ovespian and Pełczyński in [OP75] proved that every separable Banach
space admits a bounded M-basis. Later in [Peł76] Pełczyński proved that for
every $\varepsilon>0$ any separable Banach space admits an M-basis
$\\{x_{n}\\}_{n=1}^{\infty}$ with biorthogonal functionals
$\\{x_{n}^{*}\\}_{n=1}^{\infty}$ such that $\|x_{n}\|\cdot\|x_{n}^{*}\|\leq
1+\varepsilon$.

An M-basis $\\{x_{n}\\}_{n=1}^{\infty}$ such that $\|x_{n}\|=\|x_{n}^{*}\|=1$
is called an _Auerbach basis_. Every finite dimensional Banach space contains
an Auerbach basis — this statement is known as Auerbach’s lemma. However, the
question whether every separable Banach space has an Auerbach basis is a long-
standing open problem; in particular, it is not known for the space of
continuous functions on a segment $C([0,1])$ (see e.g. [GMZ16, Problem 113]).

For a sequence of non-negative numbers $(\varepsilon_{n})_{n\geq 1}$ an
M-basis $\\{x_{n}\\}_{n=1}^{\infty}$ is called _$(1+\varepsilon_{n})$
-bounded_ if $\|x_{n}\|\cdot\|x_{n}^{*}\|\leq 1+\varepsilon_{n}$; if
$\varepsilon_{n}$ tends to $0$, then it is natural to call such M-bases
_almost Auerbach_. Vershynin in [Ver99] proved that if $\sum_{n\geq
1}\varepsilon_{n}^{2}=\infty$, then every separable Banach space contains a
$(1+\varepsilon_{n})$-bounded M-basis.

Finally, let us introduce one more definition. An M-basis
$\\{x_{n}\\}_{n=1}^{\infty}$ is called _strong_ if
$x\in\overline{\operatorname{span}}\\{x_{n}^{*}(x)x_{n}\\}_{n=1}^{\infty}$ for
any $x\in X$. If $X$ is a Hilbert space, then this property is equivalent to
_hereditary completeness_ and it is related to so-called spectral synthesis of
operators on Hilbert spaces, see e.g. [Mar70]. The examples of bounded non-
strong M-bases in a separable Hilbert space can be found e.g. in [Mar70, DN76,
DNS77]. Obviously, if an M-basis is not strong, then it is not a Schauder
basis after any permutation of its elements.

###  1.2. Formulations of the results

An Auerbach basis in a Hilbert space is an orthonormal basis. It is therefore
natural to ask if $(1+\varepsilon_{n})$-bounded M-bases also have some good
properties for sufficiently small numbers $\varepsilon_{n}$. In a recent paper
[RWZ24] it is shown that it is indeed the case: if $\sum_{n\geq
1}\varepsilon_{n}<\infty$, then any (normalized) $(1+\varepsilon_{n})$-bounded
M-basis in a separable Hilbert space must be a Riesz, or unconditional, basis
(i.e., an image of an orthonormal basis under the action of a bounded
invertible operator). Besides that, it was shown that the condition
$\sum_{n\geq 1}\varepsilon_{n}<\infty$ is sharp: if this series is divergent,
then there exists a $(1+\varepsilon_{n})$-bounded M-basis which is not a Riesz
basis.

Note that Riesz bases are always Schauder bases while the converse is not true
(the examples of Schauder bases which are not Riesz bases in a Hilbert space
can be found e.g. in [AK16, Section 9.5]). When it comes to the notion of a
Schauder basis, in the paper [RWZ24] it was shown that if the sequence
$(\varepsilon_{n})_{n\geq 1}$ decays not very fast, then there exists a
$(1+\varepsilon_{n})$-bounded M-basis which is not a Schauder basis after any
permutation. However, the required assumption on this sequence is considerably
stronger than the divergence of the series $\sum_{n\geq 1}\varepsilon_{n}$. To
be more precise, it was proved that if
$\lim_{n\to\infty}n\varepsilon_{n}=\infty$, then there exists a
$(1+\varepsilon_{n})$-bounded M-basis in $\ell^{2}$ which is not a Schauder
basis in any order. This basis was constructed from finite-dimensional
“blocks” and hence it is a strong M-basis. Finally, using this finite-
dimensional construction, the method of Vershynin from the paper [Ver99] and
Dvoretzky’s theorem, the following corollary was proved in [RWZ24]: if
$\lim_{n\to\infty}n\varepsilon_{n}=\infty$ and
$\sum_{n=1}^{\infty}\varepsilon_{n}^{2}=\infty$, then in any separable Banach
space $X$ there exists a $(1+\varepsilon_{n})$-bounded M-basis which cannot be
ordered to become a Schauder basis.

Our goal is to improve the aforementioned results from [RWZ24]. In particular,
we will get rid of the condition $\lim_{n\to\infty}n\varepsilon_{n}=\infty$.
At first, we will present a simple explicit construction which proves the
following statement.

######  Theorem 1.

For any sequence of nonnegative numbers $(\varepsilon_{n})_{n\geq 1}$ such
that $\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$ there exists a
$(1+\varepsilon_{n})$-bounded M-basis in $\ell^{2}$ which is not strong.

Non-strong M-bases are not Riesz bases and moreover, as we mentioned before,
they do not form Schauder bases after any permutation.

In order to obtain a result for arbitrary Banach spaces using Dvoretzky’s
theorem, we will need the following finite-dimensional version of our
construction.

######  Theorem 2.

Suppose that $(\varepsilon_{n})_{n\geq 1}$ is a sequence of nonnegative
numbers such that $\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$. Then for any
$C>0$ there exists an integer number $N$, a finite-dimensional Hilbert space
$H$, $\dim H=N$, and a biorthogonal system $\\{(x_{n},x_{n}^{*})\\}_{n=1}^{N}$
in $H$ such that $\|x_{n}\|\cdot\|x_{n}^{*}\|\leq 1+\varepsilon_{n}$ and for
any bijection $\pi:\\{1,\ldots,N\\}\to\\{1,\ldots,N\\}$ the basis constant of
$(x_{\pi(k)})_{k=1}^{N}$ is at least $C$.

Now, using Vershynin’s construction from the paper [Ver99] and Dvoretzky’s
theorem, the following corollary can be proved in exactly the same way as it
is done in [RWZ24] — but the extra condition
$\lim_{n\to\infty}n\varepsilon_{n}=\infty$ is no longer required (note that if
$\sum_{n=1}^{\infty}\varepsilon_{n}^{2}=\infty$, then of course also
$\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$).

######  Corollary 1.

Suppose that $(\varepsilon_{n})_{n\geq 1}$ is a sequence of nonnegative
numbers such that $\sum_{n=1}^{\infty}\varepsilon_{n}^{2}=\infty$. Then in any
separable Banach space $X$ there exists a $(1+\varepsilon_{n})$-bounded
M-basis which cannot be ordered to become a Schauder basis.

Besides that, we can use Theorem 2 in order to construct a
$(1+\varepsilon_{n})$-bounded strong M-basis which is not a Schauder basis
after any permutation of its elements:

######  Corollary 2.

For any sequence of nonnegative numbers $(\varepsilon_{n})_{n\geq 1}$ such
that $\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$ there exists a
$(1+\varepsilon_{n})$-bounded strong M-basis in $\ell^{2}$ which is not a
Schauder basis after any permutation.

Indeed, the construction is straightforward. We can take an arbitrary sequence
$C_{m}$ tending to infinity. Then, using Theorem 2, at first we find the space
$H_{1}$ with $\dim H_{1}=N_{1}$ and a
$(1+\varepsilon_{n})_{n=1}^{N_{1}}$-bounded biorthogonal system in $H_{1}$
such that after any permutation its basis constant is at least $C_{1}$. After
that we use Theorem 2 once again for a sequence
$(\varepsilon_{n+N_{1}})_{n\geq 1}$ and find a space $H_{2}$ with $\dim
H_{2}=N_{2}$ and a $(1+\varepsilon_{n+N_{1}})_{n=1}^{N_{2}}$-bounded
biorthogonal system in $H_{2}$ such that after any permutation its basis
constant is at least $C_{2}$. We continue this process, and then we take the
union of the constructed systems in $\bigoplus_{m=1}^{\infty}H_{m}=\ell^{2}$.
Since our system in constructed from finite-dimensional “blocks”, it is easy
to see that it forms a strong M-basis.

##  2\. Proof of Theorem 1

Let us prove Theorem 1. First of all, we can assume that all numbers
$\varepsilon_{n}$ are strictly positive (because we can exclude all zeroes
from the sequence $(\varepsilon_{n})_{n\geq 1}$, construct the required
M-basis and then add extra coordinates to our Hilbert space and extra
orthonormal vectors to our construction). After that, replacing, if necessary,
each number $\varepsilon_{n}$ with a smaller number we can assume that
$\varepsilon_{n}\to 0$ (but still the series $\sum_{n\geq 1}\varepsilon_{n}$
is divergent). Finally, after a sufficient permutation, we can also assume
that the sequence $(\varepsilon_{n})$ is non-increasing:
$\varepsilon_{n+1}\leq\varepsilon_{n}$ for every $n$.

Now let us define by induction the following sequence of positive numbers
$(c_{n})_{n=1}^{\infty}$:

(2.1) $c_{1}=\varepsilon_{1};\quad
c_{n+1}=\frac{\varepsilon_{n+1}}{1+c_{1}+c_{2}+\ldots+c_{n}}.$

We claim that $\sum_{n=1}^{\infty}c_{n}=\infty$. Indeed, suppose that
$\sum_{n=1}^{\infty}c_{n}=A<\infty$. Then

$c_{n+1}=\frac{\varepsilon_{n+1}}{1+c_{1}+\ldots+c_{n}}\geq\frac{\varepsilon_{n+1}}{1+A}$

and therefore

$\sum_{k=1}^{\infty}\varepsilon_{k}\leq(1+A)\sum_{k=1}^{\infty}c_{k}=A(1+A)<\infty,$

which contradicts the assumption of the divergence of series $\sum_{n\geq
1}\varepsilon_{n}$.

It is not difficult to see that the sequence $(c_{n})_{n=1}^{\infty}$ is non-
increasing: for every $n\geq 1$ we have

(2.2)
$c_{n+1}=\frac{\varepsilon_{n+1}}{1+c_{1}+\ldots+c_{n}}\leq\frac{\varepsilon_{n}}{1+c_{1}+\ldots+c_{n}}\leq\frac{\varepsilon_{n}}{1+c_{1}+\ldots+c_{n-1}}=c_{n}.$

Since the sequence $(c_{n})$ is non-increasing and the series
$\sum_{n=1}^{\infty}c_{n}$ diverges, we also have

(2.3) $\displaystyle\sum_{k=1}^{\infty}c_{2k}$ $\displaystyle=\infty;$ (2.4)
$\displaystyle\sum_{l=1}^{\infty}c_{2l-1}$ $\displaystyle=\infty.$

Let us define the following numbers for $m,n\geq 1$:

(2.5) $\displaystyle\beta_{m}$ $\displaystyle=c_{m}^{1/2};$ (2.6)
$\displaystyle\alpha_{m,n}$
$\displaystyle=\begin{cases}-\beta_{m}\beta_{n},&m>n\\\ 0,&m\leq
n\end{cases}.$

Suppose now that $\\{\mathfrak{e}\\}\cup\\{e_{n}\\}_{n=1}^{\infty}$ is an
orthonormal basis in $\ell^{2}$. We define the vectors
$\\{x_{n}\\}_{n=1}^{\infty}$ and $\\{y_{n}\\}_{n=1}^{\infty}$ in our space
$\ell^{2}$ as follows:

(2.7) $\displaystyle x_{2l-1}$ $\displaystyle=e_{2l-1};\
y_{2l-1}=\beta_{2l-1}\mathfrak{e}+e_{2l-1}+\sum_{k=1}^{\infty}\alpha_{2l-1,2k}e_{2k},\
l\geq 1$ (2.8) $\displaystyle y_{2k}$ $\displaystyle=e_{2k};\qquad
x_{2k}=\beta_{2k}\mathfrak{e}+e_{2k}+\sum_{l=1}^{\infty}\alpha_{2k,2l-1}e_{2l-1},\
\ k\geq 1.$

Note that each sum in this definition contains only a finite number of nonzero
summands. We claim that the system $\\{y_{n}\\}_{n=1}^{\infty}$ is
biorthogonal to $\\{x_{n}\\}_{n=1}^{\infty}$, both these systems are complete
in $\ell^{2}$ but $\\{x_{n}\\}_{n=1}^{\infty}$ is not a strong M-basis and
$\|x_{n}\|\cdot\|y_{n}\|\leq 1+\varepsilon_{n}$ for every $n\geq 1$. These
properties imply that $\\{x_{n}\\}_{n=1}^{\infty}$ serves as an example for
Theorem 1. In what follows we will verify these properties.

###  2.1. Completeness

Let us prove that the system $\\{x_{n}\\}_{n=1}^{\infty}$ is complete in
$\ell^{2}$ (the completeness of $\\{y_{n}\\}_{n=1}^{\infty}$ can be proved in
the same way). Suppose that $h\in\ell^{2}$ and $\langle h,x_{n}\rangle=0$ for
every $n\geq 1$. Then in particular $\langle h,x_{2l-1}\rangle=\langle
h,e_{2l-1}\rangle=0$. Therefore, the relation $\langle h,x_{2k}\rangle=0$ is
equivalent to $\langle h,e_{2k}\rangle=-\beta_{2k}\langle
h,\mathfrak{e}\rangle$. If we now denote $c=\langle h,\mathfrak{e}\rangle$, we
see that

$\sum_{k=1}^{\infty}\langle
h,e_{2k}\rangle^{2}=c^{2}\sum_{k=1}^{\infty}\beta_{2k}^{2}=c^{2}\sum_{k=1}^{\infty}c_{2k}.$

The latter series diverges (see formula (2.3)), hence, since $h\in\ell^{2}$ we
infer that $c$ must be equal to $0$ — but then $\langle
h,\mathfrak{e}\rangle=\langle h,e_{n}\rangle=0$ for every $n\geq 1$ which
means that $h=0$ in $\ell^{2}$. This proves the completeness of the system
$\\{x_{n}\\}_{n=1}^{\infty}$ and, as we mentioned, completeness of the system
$\\{y_{n}\\}_{n=1}^{\infty}$ follows in a similar way.

###  2.2. Biorthogonality

The relations $\langle x_{2l-1},y_{2k}\rangle=0$, $\langle
x_{2l-1},y_{2m-1}\rangle=\delta_{lm}$ and $\langle
x_{2k},y_{2m}\rangle=\delta_{km}$ are trivial. We only need to check that
$\langle x_{2k},y_{2l-1}\rangle=0$. But the direct computation shows that
$\langle
x_{2k},y_{2l-1}\rangle=\beta_{2l-1}\beta_{2k}+\alpha_{2l-1,2k}+\alpha_{2k,2l-1}$.
This quantity equals to $0$ by the definition of the numbers $\alpha_{m,n}$
(see formula (2.6)).

###  2.3. The basis is not strong

We have just shown that $\\{x_{n}\\}_{n=1}^{\infty}$ is an M-basis in
$\ell^{2}$ and $\\{y_{n}\\}_{n=1}^{\infty}$ is its biorthogonal system.
However, it is straightforward to see that this M-basis is not strong. Indeed
$\langle\mathfrak{e},y_{n}\rangle\neq 0$ only for odd numbers $n$ and hence

$\mathfrak{e}\not\in\overline{\operatorname{span}}\\{e_{2l-1}\\}_{l=1}^{\infty}=\overline{\operatorname{span}}\\{\langle\mathfrak{e},y_{n}\rangle
x_{n}\\}_{n=1}^{\infty}.$

###  2.4. The norm estimate

Finally, it remains only to check that $\|x_{n}\|\cdot\|y_{n}\|\leq
1+\varepsilon_{n}$ for all $n\geq 1$. Let us check it for even numbers $n=2k$
(the computation for odd numbers is completely similar). In this case
$\|y_{2k}\|=1$ and

$\|x_{2k}\|^{2}=1+\beta_{2k}^{2}+\sum_{l=1}^{\infty}\alpha_{2k,2l-1}^{2}=1+\beta_{2k}^{2}+\sum_{l=1}^{k}\beta_{2k}^{2}\beta_{2l-1}^{2}=1+c_{2k}\Big{(}1+\sum_{l=1}^{k}c_{2l-1}\Big{)}\\\
\leq 1+c_{2k}\Big{(}1+\sum_{n=1}^{2k-1}c_{n}\Big{)}=1+\varepsilon_{2k}.$

Using this inequality, we get that
$\|x_{2k}\|\leq(1+\varepsilon_{2k})^{1/2}\leq 1+\varepsilon_{2k}$, and we are
done.

##  3\. Proof of Theorem 2

As in the proof of Theorem 1, we can assume that the numbers $\varepsilon_{n}$
are positive and the sequence $(\varepsilon_{n})_{n\geq 1}$ tends to $0$.
Besides, we can assume that $\varepsilon_{n+1}\leq\varepsilon_{n}$ for all
$n\geq 1$. Indeed, note that if we construct a Hilbert space $H$ with a basis
$(x_{n})_{n=1}^{N}$ in it which satisfies the conditions of Theorem 2 for non-
increasing permutation of the numbers $(\varepsilon_{n})_{n\geq 1}$, then, for
a given arbitrary order of $(\varepsilon_{n})_{n\geq 1}$, we can consider the
Hilbert space $\widetilde{H}$ of sufficiently large dimension $K$, fix an
orthonormal basis $\\{w_{n}\\}_{n=1}^{K}$ in it, and take the space
$\widetilde{H}\oplus H$ with a sufficient permutation of the basis
$\\{w_{n}\oplus 0\\}_{n=1}^{K}\cup\\{0\oplus x_{n}\\}_{n=1}^{N}$ in it.

Let us define the numbers $\beta_{n}$ and $\alpha_{m,n}$ in the same way as in
the proof of Theorem 1 (see formulas (2.5) and (2.6)). Recall that, due to our
assumptions, as before, the sequence $(\beta_{n})$ is non-increasing (see the
formulas (2.2) and (2.5)). The formulas (2.3) and (2.4) can be rewritten as
follows:

(3.1) $\displaystyle\sum_{k=1}^{\infty}\beta_{2k}^{2}$ $\displaystyle=\infty;$
(3.2) $\displaystyle\sum_{l=1}^{\infty}\beta_{2l-1}^{2}$
$\displaystyle=\infty.$

Hence we can fix a number $L$ such that

(3.3) $\sum_{l=1}^{L}\beta_{2l-1}^{2}\geq 64C^{2}.$

After that we can find a number $M>L$ such that

(3.4)
$\sum_{k=1}^{L}\beta_{2k}^{2}\leq\frac{1}{8}\sum_{k=1}^{M}\beta_{2k}^{2}.$

We can also assume that

(3.5) $\sum_{k=1}^{M}\beta_{2k}^{2}>1$

We put $N=2M$.

Consider, as before, the space $\ell^{2}$ with an orthonormal basis
$\\{\mathfrak{e}\\}\cup\\{e_{n}\\}_{n=1}^{\infty}$. Recall the formulas (2.7)
and (2.8) which define the biorthogonal system
$\\{(x_{n},y_{n})\\}_{n=1}^{\infty}$ in $\ell^{2}$. For the reader’s
convenience, we rewrite them here is the following form:

(3.6) $\displaystyle x_{2l-1}$ $\displaystyle=e_{2l-1};\
y_{2l-1}=\beta_{2l-1}\mathfrak{e}+e_{2l-1}-\sum_{k=1}^{l-1}(\beta_{2l-1}\beta_{2k})e_{2k},\
l\geq 1;$ (3.7) $\displaystyle y_{2k}$ $\displaystyle=e_{2k};\ \ \ \ \
x_{2k}=\beta_{2k}\mathfrak{e}+e_{2k}-\sum_{l=1}^{k}(\beta_{2k}\beta_{2l-1})e_{2l-1},\
\ k\geq 1.$

Now we take $H=\mathrm{span}\\{x_{1},x_{2},\ldots,x_{2M}\\}\subset\ell^{2}$
and denote by $P$ an orthogonal projection onto $H$ in $\ell^{2}$. Then, if we
put $x_{n}^{*}=Py_{n}$, we get that the system $(x_{n}^{*})_{n=1}^{2M}$ is
biorthogonal to $(x_{n})_{n\geq 1}$ in $H$. Besides that, in the proof of
Theorem 1 we checked that $\|x_{n}\|\cdot\|y_{n}\|\leq 1+\varepsilon_{n}$ for
every $n$, and, since orthogonal projection does not increase the norm of the
vector, $\|x_{n}\|\cdot\|x_{n}^{*}\|\leq 1+\varepsilon_{n}$. It remains to
check that for any permutation $\pi:\\{1,\ldots,2M\\}\to\\{1,\ldots,2M\\}$ the
basis constant of $(x_{\pi(n)})_{n=1}^{2M}$ is at least $C$.

We denote

$B=\sum_{k=1}^{M}\beta_{2k}^{2}.$

Note that $B>1$ by (3.5). Now consider the following coefficients:

(3.8) $\displaystyle\gamma_{2k}$ $\displaystyle=\frac{\beta_{2k}}{B},\ 1\leq
k\leq M;$ (3.9) $\displaystyle\gamma_{2l-1}$
$\displaystyle=\frac{\beta_{2l-1}}{B}\Big{(}\sum_{k=l}^{M}\beta_{2k}^{2}\Big{)},\
1\leq l\leq M.$

Finally, let us consider the following element of $H$:

(3.10) $z=\sum_{n=1}^{2M}\gamma_{n}x_{n}.$

Let us compute $\|z\|$. Obviously,
$z\in\mathrm{span}\\{\mathfrak{e},e_{1},e_{2},\ldots e_{2M}\\}$ in $\ell^{2}$.
A direct computation shows that for $1\leq k,l\leq M$

$\langle z,\mathfrak{e}\rangle=1;\ \langle z,e_{2l-1}\rangle=0;\ \langle
z,e_{2k}\rangle=\frac{\beta_{2k}}{B}.$

Hence

$\|z\|=\Big{(}1+\sum_{k=1}^{M}\frac{\beta_{2k}^{2}}{B^{2}}\Big{)}^{1/2}=(1+1/B)^{1/2}\leq
2.$

Let us now fix a permutation $\pi:\\{1,\ldots,2M\\}\to\\{1,\ldots,2M\\}$. If
we find a number $t$ such that

$\Big{\|}\sum_{m=1}^{t}\gamma_{\pi(m)}x_{\pi(m)}\Big{\|}\geq 2C,$

then we are done.

There exists a unique number $t$ such that

(3.11) $\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}\geq\frac{B}{2},\quad\sum_{\\{m\leq
t-1:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}<\frac{B}{2}.$

Since for every $k\geq 1$ we have $\beta_{2}\geq\beta_{2k}$, these relations
imply

(3.12) $\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}<\frac{B}{2}+\beta_{\pi(t)}^{2}\leq\frac{B}{2}+\beta_{2}^{2}.$

We put

$w=\sum_{m=1}^{t}\gamma_{\pi(m)}x_{\pi(m)}\in H\subset\ell^{2}.$

We will prove now that $\|w\|\geq 2C$.

Let us compute the quantity $\langle w,e_{2l-1}\rangle$ for $l=1,2,\ldots,L$.
Note that $\langle x_{n},e_{2l-1}\rangle\neq 0$ if either $n$ is even and
$n\geq 2l$ or $n=2l-1$. Hence, if
$2l-1\not\in\\{\pi(1),\pi(2),\ldots,\pi(t)\\}$, then, using the formulas (3.7)
and (3.8), we get

(3.13) $\langle w,e_{2l-1}\rangle=-\frac{1}{B}\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2,\,\pi(m)\geq
2l\\}}\beta_{\pi(m)}^{2}\beta_{2l-1}.$

In this case, using the formulas (3.4) and (3.11), we can estimate this
quantity as follows:

(3.14) $|\langle
w,e_{2l-1}\rangle|\geq\frac{\beta_{2l-1}}{B}\Big{(}\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}-\sum_{k=1}^{l-1}\beta_{2k}^{2}\Big{)}\\\
\geq\frac{\beta_{2l-1}}{B}\Big{(}\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}-\sum_{k=1}^{L}\beta_{2k}^{2}\Big{)}\geq\frac{\beta_{2l-1}}{B}\Big{(}\frac{B}{2}-\frac{B}{8}\Big{)}\geq\frac{\beta_{2l-1}}{4}.$

Now we will compute $\langle w,e_{2l-1}\rangle$ in the case when
$2l-1\in\\{\pi(1),\pi(2),\ldots,\pi(t)\\}$. In this case we should add
$\gamma_{2l-1}$ to the right-hand side of the formula (3.13) and get

$\langle w,e_{2l-1}\rangle=-\frac{1}{B}\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2,\,\pi(m)\geq
2l\\}}\beta_{\pi(m)}^{2}\beta_{2l-1}+\frac{\beta_{2l-1}}{B}\Big{(}\sum_{k=l}^{M}\beta_{2k}^{2}\Big{)}\\\
=\frac{\beta_{2l-1}}{B}\sum_{\\{m>t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2,\,\pi(m)\geq
2l\\}}\beta_{\pi(m)}^{2}.$

Using the formulas (3.12) and (3.4), in this case we can write:

(3.15) $|\langle
w,e_{2l-1}\rangle|\geq\frac{\beta_{2l-1}}{B}\Big{(}B-\sum_{\\{m\leq
t:\,\pi(m)\mathrel{\text{\vbox{\hbox{.}\hbox{.}\hbox{.}}}}2\\}}\beta_{\pi(m)}^{2}-\sum_{k=1}^{{l-1}}\beta_{2k}^{2}\Big{)}\\\
\geq\frac{\beta_{2l-1}}{B}\Big{(}B-\frac{B}{2}-\beta_{2}^{2}-\sum_{k=1}^{L}\beta_{2k}^{2}\Big{)}\geq\frac{\beta_{2l-1}}{B}\Big{(}\frac{B}{2}-\frac{B}{8}-\frac{B}{8}\Big{)}=\frac{\beta_{2l-1}}{4}.$

Summing up, the formulas (3.14) and (3.15) imply that in either case $|\langle
w,e_{2l-1}\rangle|\geq\frac{\beta_{2l-1}}{4}$ for $l=1,2,\ldots L$. Hence, by
(3.3),

$\|w\|\geq\Big{(}\sum_{l=1}^{L}\langle
w,e_{2l-1}\rangle^{2}\Big{)}^{1/2}\geq\frac{1}{4}\Big{(}\sum_{l=1}^{L}\beta_{2l-1}^{2}\Big{)}^{1/2}\geq
2C,$

and the theorem is proved.

## Acknowledgements

The author is grateful to Nir Lev for fruitful discussions.

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