THE -PROPERTY OF A BANACH SPACE ALONG A FILTER

TOMASZ KANIA; JAROSŁAW SWACZYNA

doi:10.1017/jsl.2026.10236

THE $\pi $-PROPERTY OF A BANACH SPACE ALONG A FILTER

Part of: Normed linear spaces and Banach spaces; Banach lattices Set theory

Published online by Cambridge University Press: 13 July 2026

TOMASZ KANIA

and

JAROSŁAW SWACZYNA

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TOMASZ KANIA: Affiliation:
INSTITITUTE OF MATHEMATICS CZECH ACADEMY OF SCIENCES PRAGUE INSTITITUTE OF MATHEMATICS JAGIELLONIAN UNIVERSITY KRAKOW POLAND E-mail: tomasz.marcin.kania@gmail.com
JAROSŁAW SWACZYNA*: Affiliation:
INSTITITUTE OF MATHEMATICS LODZ UNIVERSITY OF TECHNOLOGY POLAND
*: E-mail: jaroslaw.swaczyna@p.lodz.pl

Article contents

Abstract
Introduction
Preliminaries
Proof of Theorem
Proof of Theorems and
Problems
Funding
References

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Abstract

We examine the analyticity of the class of separable Banach spaces possessing the $\pi $-property, defined in terms of convergence along a filter. Our results establish that this class is $\Sigma ^1_3$ whenever the underlying filter is analytic (as a subset of the Cantor set $\Delta $). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is $\Sigma ^1_2$ with respect to any admissible Polish topology on the family of closed subspaces of $C(\Delta )$.

Keywords

filter bases in Banach spaces polish spaces of separable Banach spaces

MSC classification

Primary: 03E15: Descriptive set theory 46B15: Summability and bases

Information

Type: Article
Information: The Journal of Symbolic Logic , First View , pp. 1 - 15

DOI: https://doi.org/10.1017/jsl.2026.10236 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1 Introduction

A Banach space is said to have the $\pi $ -property if there exists a uniformly bounded net of finite-rank projections acting on the space that converges to the identity map in the strong operator topology (SOT), i.e., pointwise. For separable Banach spaces, the $\pi $ -property is strongly connected to the existence of a finite-dimensional decomposition (FDD): the presence of an FDD ensures the $\pi $ -property, yet whether the converse holds remains an open question [Reference Hájek and Medina10, p. 3]. Notably, the stronger metric $\pi $ -property does imply the existence of an FDD.

To further explore this relationship, we propose extending the framework to encompass a broader class of spaces. Specifically, we replace convergence in the SOT with convergence along a filter, while maintaining the setting of the SOT. This generalisation not only sheds new light on the $\pi $ -property but also introduces a concept of intrinsic interest. To lay the groundwork for this study, we formally present the central definition underpinning this note that is expressed in terms of filter convergence. (Let ${\mathscr {F}}$ be a filter on $\mathbb {N}$ , and let X be a Banach space. A sequence $(x_n)_{n=1}^\infty $ in X is said to converge to $x \in X$ along ${\mathscr {F}}$ if

$$\begin{align*}\lim_{n, {\mathscr{F}}} \|x_n - x\| = 0, \end{align*}$$

meaning that for every $\varepsilon> 0$ , there exists $A \in {\mathscr {F}}$ such that $\|x_n - x\| < \varepsilon $ for all $n \in A$ .)

Definition 1.1 ( ${\mathscr {F}}$ - $\pi $ -property).

Let ${\mathscr {F}}$ be a filter on $\mathbb {N}$ . A (separable) Banach space X has the ${\mathscr {F}}$ - $\pi $ -property if there exists a sequence of finite-rank projections $(P_n)_{n=1}^\infty $ on X such that, for every $x \in X$ ,

$$\begin{align*}\lim_{n, {\mathscr{F}}} \|P_n x - x\| = 0. \end{align*}$$

In this case, the sequence $(P_n)_{n=1}^\infty $ is called an ${\mathscr {F}}$ - $\pi $ -basis of X. An ${\mathscr {F}}$ - $\pi $ -basis with respect to the Fréchet filter is simply referred to as a $\pi $ -basis.

For the sake of avoiding pathologies, we have stated the definition for filters on $\mathbb N$ thereby limiting ourselves to separable Banach spaces. Hereinafter, all filters considered contain the Fréchet filter, that is, the filter of cofinite subsets of $\mathbb N$ . It is readily seen that for the Fréchet filter, Definition 1.1 renders the familiar $\pi $ -property.

Banach spaces that have filter bases having continuous basis projections constitute paradigmatic examples of spaces with the ${\mathscr {F}}$ - $\pi $ -property. Indeed, suppose that $(e_n)_{n=1}^\infty $ is an ${\mathscr {F}}$ -basis for a Banach space X. This means that for every $x\in X$ there exists a unique sequence of scalars $(e_n^*(x))_{n=1}^\infty $ such that

• for every $n\in \mathbb N$ , the assignment $y\mapsto e_n^*(y)$ is continuous;
• $\lim \limits _{n, {\mathscr {F}}}\sum _{j=1}^n e_j^*(x)e_j =x$ .

The finite-rank operators $(P_n)_{n=1}^\infty $ given by $P_n x = \sum _{j=1}^n e_j^*(x) e_j$ (basis projections) witness the ${\mathscr {F}}$ - $\pi $ -property of X.

Kochanek [Reference Kochanek14] demonstrated that if ${\mathscr {F}}$ is countably generated (or more generally, generated by fewer than $\mathfrak {p}$ sets, where $\mathfrak {p}$ denotes the pseudo-intersection number), that is, there exists a countable family (or a family of cardinality less than $\mathfrak {p}$ ) $\mathscr {B} \subset {\mathscr {F}}$ such that for every $ A \in {\mathscr {F}} $ , there exists $ B \in \mathscr {B} $ with $ B \subseteq A $ , then the associated projections $(P_n)_{n=1}^\infty $ are uniformly bounded on some set in the filter, and the continuity assumption in the definition of an ${\mathscr {F}}$ -basis becomes redundant.

Moreover, the present authors have recently shown that the continuity assumption in the definition of an ${\mathscr {F}}$ -basis is redundant for projective filters under proper set-theoretic assumptions [Reference Kania and Swaczyna12] and even for analytic filters in ZFC (jointly with de Rancourt [Reference de Rancourt, Kania and Swaczyna15]).

The proof techniques from [Reference Kochanek14] inspired Avilés et al. [Reference Avilés, Kadets, Pérez and Solecki1] to identify the class of filters possibly generated by $\mathfrak {c}$ -sized sets for which Kochanek’s argument remains valid. A filter ${\mathscr {F}}$ is termed Baire if, for every complete metric space $ X $ and every family $\{X_A \colon A \in {\mathscr {F}}\}$ of nowhere-dense subsets of $ X $ satisfying $ X_A \subseteq X_B $ whenever $ A, B \in {\mathscr {F}} $ and $ A \subseteq B $ , it holds that $\bigcup _{A \in {\mathscr {F}}} X_A \neq X$ .

Under Martin’s Axiom, the classes of filters that are Baire with respect to all complete metric spaces and filters generated by fewer than continuum-many sets coincide. However, there exist models of ZFC that admit filters generated by exactly $\mathfrak {c}$ -sized sets that are Baire [Reference Avilés, Kadets, Pérez and Solecki1, Section 5]. On the other hand, analytic filters that are not countably generated fail to be Baire with respect to some Polish space [Reference Avilés, Kadets, Pérez and Solecki1, Proposition 4.1], which constrains the range of spaces with the ${\mathscr {F}}$ - $\mathcal {\pi }$ -property derived from ${\mathscr {F}}$ -bases.

An FDD for a Banach space X is a sequence of finite-dimensional subspaces $(F_n)_{n=1}^\infty $ such that every $x \in X$ admits a unique representation

$$\begin{align*}x = \sum_{n=1}^\infty x_n, \quad \text{with} \quad x_n \in F_n \ \text{for all} \ n \in \mathbb{N}. \end{align*}$$ x equals sigma summation Underscript n equals 1 Overscript infinity Endscripts x Subscript n Baseline comma with x Subscript n Baseline element of upper F Subscript n Baseline for all n element of double struck upper N period

Let us formulate the following question, which encapsulates our motivation for investigating spaces with the ${\mathscr {F}}$ - $\mathcal {\pi }$ -property.

Question 1.2. Suppose that a separable Banach space has the ${\mathscr {F}}$ - $\mathcal {\pi }$ -property with respect to a certain filter ${\mathscr {F}}$ . Does $ X $ have an FDD?

The purpose of this note is to evaluate the complexity of spaces with the ${\mathscr {F}}$ - $\mathcal {\pi }$ -property within the framework of analytic filters. This investigation is inspired by a similar result of Ghawadrah [Reference Ghawadrah8], which concerns spaces with the usual $\pi $ -property.

To state our result, we introduce the set $\mathsf {SB}_{{\mathscr {F}}\text {-}\pi }$ , a subset of $\mathsf {SB}$ (formally defined in the following section) consisting of Banach spaces that exhibit the ${\mathscr {F}}$ - $\mathcal {\pi }$ -property.

Theorem A. Let ${\mathscr {F}}$ be an analytic (or $\Sigma ^1_{s}$ ) filter on $\mathbb N$ . Then:

• The set $\mathsf {SB}_{{\mathscr {F}}\text {-}\pi }$ is $\Sigma _3^1$ ( $\Sigma _{s+2}^1$ ) in $\mathsf {SB}$ .
• If ${\mathscr {F}}$ is countably generated, the set $\mathsf {SB}_{{\mathscr {F}}\text {-}\pi }$ is $\Sigma ^1_2$ in $\mathsf {SB}$ .

For the Fréchet filter ${\mathscr {F}}_{{\mathrm {Fr}}}$ , that is, in the case of the usual $\pi $ -property, Ghawadrah [Reference Ghawadrah8] proved that $\mathsf {SB}_{{\mathscr {F}}_{\mathrm {Fr}}\text {-}\pi }$ is $\Sigma ^0_6$ . Even though neither Ghawadrah’s identification of the Borel class of $\mathsf {SB}_{{\mathscr {F}}_{\mathrm {Fr}}\text {-}\pi }$ nor our identification of $\mathsf {SB}_{{\mathscr {F}}\text {-}\pi }$ is proved to be optimal, the increase in Borel complexity by one appears to be inevitable. This suggests a negative answer to Question 1.2, should the $\pi $ -property be equivalent to the existence of an FDD for separable Banach spaces (see [Reference Casazza2, Theorem 6.4]).

Note also that the family $\mathsf {SB}_{{\mathscr {F}}\text {-}\pi }$ may depend on the particular choice of the filter ${\mathscr {F}}$ . The following theorem asserts that a certain reduction to an analytic filter is possible.

Theorem B. Let ${\mathscr {F}}$ be a filter on $\mathbb {N}$ . Suppose that X is a Banach space with the ${\mathscr {F}}$ - $\pi $ -property, and let $(P_n)_{n=1}^\infty $ witness this property. Then there exists an analytic filter ${\mathscr {F}}' \subset {\mathscr {F}}$ such that $(P_n)_{n=1}^\infty $ also witnesses the ${\mathscr {F}}'$ - $\pi $ -property of X.

It is worth noting that Theorem B does not render the consideration of higher projective (and beyond) complexities superfluous, as we do not assert that the choice of ${\mathscr {F}}'$ can be made uniformly for all $X \in \mathsf {SB}$ .

Theorem C. Let $\mathsf {SB}_{\operatorname {Filter-}\pi }$ denote the class of all $X \in \mathsf {SB}$ which possess the ${\mathscr {F}}$ - $\pi $ -property for some filter ${\mathscr {F}}$ . Then $\mathsf {SB}_{\operatorname {Filter-}\pi }$ is $\Sigma ^1_3$ within $\mathsf {SB}$ .

2 Preliminaries

Our notation concerning Banach spaces is standard. We work with Banach spaces over real or complex scalar field. All operators are assumed to be bounded and linear. A projection is an idempotent operator. We denote the identity operator on a normed space F by $I_F$ , and the closed unit ball of a Banach space X by $B_X=\{x\in X\colon \|x\|\leqslant 1\}$ .

By the Banach–Mazur theorem, every separable Banach space embeds isometrically into $C(\Delta )$ , the space of continuous functions on the Cantor set. The space $\mathsf {SB}$ of all closed (linear) subspaces of $C(\Delta )$ is a standard Borel space when furnished with the Effros–Borel $\sigma $ -algebra that is generated by the sets $E^+(U) = \{X\in \mathsf {SB}\colon X\cap U\neq \emptyset \}$ , where U is a non-empty, open subset of $C(\Delta )$ . There are numerous Polish topologies $\mathsf {SB}$ whose Borel $\sigma $ -algebra coincides with the Effros–Borel $\sigma $ -algebra, albeit no canonical one.

Following Godefroy and Saint-Raymond [Reference Godefroy and Saint-Raymond9], we say that a Polish topology $\tau $ on $\mathsf {SB}$ is admissible if it satisfies the following conditions:

• For every open set $U \subseteq C(\Delta )$ , the set $E^+(U)$ belongs to $\tau $ .
• There exists a subbasis $\mathcal {B}$ of $\tau $ such that every $U \in \mathcal {B}$ can be expressed as a countable union of sets of the form $E^+(U) \setminus E^+(V)$ , where U and V are open subsets of $C(\Delta )$ .

A natural example of an admissible topology is the Wijsman topology, where $X_n \to X$ in $\mathsf {SB}$ if and only if $d(x, X_n) \to d(x, X)$ for every $x \in C(\Delta )$ . Recall that a subset of a Polish space is called analytic if it is the continuous image of a Borel set in a Polish space.

We will also make use of the following result. For a detailed study of the descriptive set-theoretic complexity of isometry classes, we refer the reader to [Reference Cúth, Doležal, Doucha and Kurka4, Reference Cúth, Doležal, Doucha and Kurka5] that build upon the results from [Reference Godefroy and Saint-Raymond9]. Note that since all codings used in those papers agree on projective classes, our results remain true in the context of all those papers.

Theorem 2.1 [Reference Godefroy and Saint-Raymond9, Theorem 4.1].

There exists a sequence of continuous maps $(g_i)_{i=1}^\infty $ , where $g_i \colon \mathsf {SB} \to C(\Delta )$ , such that for every $X \in \mathsf {SB}$ ,

$$\begin{align*}X = \overline{\{g_i(X)\colon i \in \mathbb{N}\}}. \end{align*}$$

As a consequence of the above theorem we get the following.

Corollary 2.2. There exists a sequence of Borel maps $(f_i)_{i=1}^\infty $ , where $f_i \colon \mathsf {SB} \to C(\Delta )$ , such that for every $X \in \mathsf {SB}$ ,

$$\begin{align*}X = \overline{\{f_i(X)\colon i \in \mathbb{N}\}} \end{align*}$$

and for every infinite-dimensional X vectors $f_i(X)$ are linearly independent.

Proof. Consider the functions $g_i$ provided by Theorem 2.1, and let $(U_n)_{n=1}^\infty $ be a countable base for $C(\Delta )$ . By [Reference Godefroy and Saint-Raymond9, Corollary 4.2], the set of finite-dimensional spaces is Borel in $\mathsf {SB}$ . Consequently, we may define the functions $f_i$ separately for finite- and infinite-dimensional spaces without violating Borel measurability.

For finite-dimensional spaces, we set $f_i(X) = g_i(X)$ whenever $\dim X$ is finite. For infinite-dimensional spaces $X \in \mathsf {SB}$ , we define the values $f_i(X)$ inductively. Fix $n \in \mathbb {N}$ and assume that $f_i(X)$ has already been defined for all $i < n$ . We then define

$$\begin{align*}m_n = \min \left\{ j \in \mathbb{N} \colon g_j(X) \in U_n \wedge g_j(X) \text{ is independent of } \{ f_i(X) \colon i < n \} \right\}, \end{align*}$$

whenever $U_n \cap X \neq \emptyset $ , and

$$\begin{align*}m_n = \min \left\{ j \in \mathbb{N} \colon g_j(X) \text{ is independent of } \{ f_i(X) \colon i < n \} \right\}, \end{align*}$$

if $U_n \cap X = \emptyset $ . The assumptions on the dimension of X and the density of the set $\{ g_i(X) \colon i \in \mathbb {N} \}$ in X ensure that this construction is well-defined.

It is clear that the vectors $f_n(X)$ remain linearly independent whenever $\dim X = \mathfrak {c}$ . The remaining claims follow straightforwardly. In particular, for every $n \in \mathbb {N}$ , there exists a partition of $\mathsf {SB}$ into Borel sets $(B_i)_{i=1}^\infty $ such that for each $i \in \mathbb {N}$ , there exists some $j \in \mathbb {N}$ satisfying $f_n |_{B_i} = g_j |_{B_i}$ .

2.1 Finite-dimensional structures

Let us begin with the observation that uniformly bounded ${\mathscr {F}}$ - $\pi $ -bases closely resemble classical $\pi $ -bases.

Proposition 2.3. Let X be a separable Banach space, and let ${\mathscr {F}}$ be a filter on $\mathbb {N}$ containing the Fréchet filter. Suppose that $(P_n)_{n=1}^\infty $ is an ${\mathscr {F}}$ - $\pi $ -basis of X which is uniformly bounded; that is, there exists $M> 0$ such that $\|P_n\| \leqslant M$ for all $n \in \mathbb {N}$ . Then $(P_n)_{n=1}^\infty $ admits a subsequence which is a $\pi $ -basis of X.

Proof. Since X is separable, there exists a countable dense subset $D = \{x_1, x_2, \dots \} \subseteq X$ . For each $k \in \mathbb {N}$ and $\varepsilon> 0$ , the filter convergence $\lim _{n, {\mathscr {F}}} P_n x_k = x_k$ implies that the set

$$\begin{align*}A_{k, \varepsilon} = \{ n \in \mathbb{N} : \|P_n x_k - x_k\| < \varepsilon \} \end{align*}$$

belongs to ${\mathscr {F}}$ . Since ${\mathscr {F}}$ is a filter, it is closed under finite intersections. Thus, for each $m \in \mathbb {N}$ , the set

$$\begin{align*}B_m = \bigcap_{k=1}^m A_{k, \varepsilon/m} \end{align*}$$ upper B Subscript m Baseline equals intersection Underscript k equals 1 Overscript m Endscripts upper A Subscript k comma epsilon divided by m

also lies in ${\mathscr {F}}$ . Since every filter contains the Fréchet filter, each $B_m$ is infinite.

We construct a subsequence $(n_m)_{m=1}^\infty $ inductively as follows:

• Choose $n_1 \in B_1$ .
• For $m \geqslant 2$ , choose $n_m \in B_m$ such that $n_m> n_{m-1}$ .

This construction is possible because each $B_m$ is infinite. By construction, for each $m \in \mathbb {N}$ and for all $k = 1, 2, \dots , m$ , we have

(1) $$ \begin{align} \|P_{n_m} x_k - x_k\| < \frac{\varepsilon}{m}. \end{align} $$

Now, fix $k \in \mathbb {N}$ . For all $m \geqslant k$ , equation (1) holds true. Since $\varepsilon> 0$ was arbitrary, taking the limit as $m \to \infty $ gives

$$\begin{align*}\lim_{m \to \infty} P_{n_m} x_k = x_k. \end{align*}$$

This holds for all $k \in \mathbb {N}$ , so the subsequence $(P_{n_m})$ converges pointwise to the identity operator on the dense set D. Now, let $x \in X$ and $\varepsilon> 0$ be given. Since D is dense in X, there is $x_k \in D$ such that

$$\begin{align*}\|x - x_k\| < \frac{\varepsilon}{3(M + 1)}. \end{align*}$$

Since $P_{n_m} x_k \to x_k$ , there exists $M_k \in \mathbb {N}$ such that for all $m \geqslant M_k$ ,

$$\begin{align*}\|P_{n_m} x_k - x_k\| < \frac{\varepsilon}{3}. \end{align*}$$

Therefore, for $m \geqslant M_k$ , we have

$$ \begin{align*} \|P_{n_m} x - x\| &\leqslant \|P_{n_m} x - P_{n_m} x_k\| + \|P_{n_m} x_k - x_k\| + \|x_k - x\| \\ &\leqslant \|P_{n_m}\| \cdot \|x - x_k\| + \|P_{n_m} x_k - x_k\| + \|x_k - x\| \\ &\leqslant M \cdot \frac{\varepsilon}{3(M + 1)} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3(M + 1)} \\ &< \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon. \end{align*} $$

StartLayout 1st Row 1st Column StartMetric upper P Subscript n Sub Subscript m Subscript Baseline x minus x EndMetric 2nd Column less than or slanted equals StartMetric upper P Subscript n Sub Subscript m Subscript Baseline x minus upper P Subscript n Sub Subscript m Subscript Baseline x Subscript k Baseline EndMetric plus StartMetric upper P Subscript n Sub Subscript m Subscript Baseline x Subscript k Baseline minus x Subscript k Baseline EndMetric plus StartMetric x Subscript k Baseline minus x EndMetric 2nd Row 1st Column Blank 2nd Column less than or slanted equals StartMetric upper P Subscript n Sub Subscript m Subscript Baseline EndMetric dot StartMetric x minus x Subscript k Baseline EndMetric plus StartMetric upper P Subscript n Sub Subscript m Subscript Baseline x Subscript k Baseline minus x Subscript k Baseline EndMetric plus StartMetric x Subscript k Baseline minus x EndMetric 3rd Row 1st Column Blank 2nd Column less than or slanted equals upper M dot StartFraction epsilon Over 3 left parenthesis upper M plus 1 right parenthesis EndFraction plus StartFraction epsilon Over 3 EndFraction plus StartFraction epsilon Over 3 left parenthesis upper M plus 1 right parenthesis EndFraction 4th Row 1st Column Blank 2nd Column less than StartFraction epsilon Over 3 EndFraction plus StartFraction epsilon Over 3 EndFraction plus StartFraction epsilon Over 3 EndFraction equals epsilon period EndLayout

Thus, for each $x \in X$ , we have shown that $\lim _{m \to \infty } P_{n_m} x = x$ . Therefore, the subsequence $(P_{n_m})$ is a $\pi $ -basis for X.

We shall require the following perturbation lemma [Reference Johnson, Rosenthal and Zippin11, Lemma 2.4].

Lemma 2.4. Let X be a Banach space and let $E\subseteq X$ be a finite-dimensional subspace. Let $T\colon X\to E$ be a surjective operator. If $F\subseteq X$ is a subspace with $k = \dim F \leqslant \dim E = n$ such that $\|T|_F - I_F\| < \varepsilon < 1$ , where $\varepsilon \cdot k / (1-\varepsilon ) < 1$ , then

i) there is an operator S from X onto an n-dimensional subspace of X such that $S|_F = I_F$ , $\|S-T\| < \varepsilon \cdot k\cdot \|T\| / (1-\varepsilon ) $ , and $S^*[X^*] = T^*[X^*]$ ;
ii) if T is a projection, S may also be arranged to be such, in which case we have $\|S|_{T[X]} - I_{T[X]}\| < \varepsilon \cdot k / (1-\varepsilon )$ .

Corollary 2.5. Let $ X $ be a Banach space, and let $ P \colon X \to X $ be a finite-rank projection. Suppose $ D = \{x_i \colon i \in \mathbb {N}\} $ is a dense subset of $ X $ . Define $ X_k $ as the linear span of $ \{x_1, \ldots , x_k\} $ for $ k \in \mathbb {N} $ . Then, for every $ \varepsilon \in (0, 1) $ , there exist a projection $ P^\prime $ and $ m \in \mathbb {N} $ such that:

• $ \|P - P^\prime \| \leqslant \varepsilon \cdot k \cdot \|P\| \cdot (1 - \varepsilon )^{-1} $ , where $ k = \dim \operatorname {im} P $ ;
• $ \operatorname {im} P^\prime \subseteq X_m $ ;
• $ \|P^\prime |_{\operatorname {im} P} - I_{\operatorname {im} P}\| < \varepsilon \cdot \dim \operatorname {im} P / (1 - \varepsilon ) $ .

For the sake of completeness, we sketch the proof of Corollary 2.5.

Proof. Put $ k=\dim \operatorname {im} P $ . Choose $0<\eta <\varepsilon $ so small that $\eta k/(1-\eta )<1$ . By the density of D, there is $m\in \mathbb N$ and a subspace $F\subset X_m$ with $\dim F=k$ such that

$$\begin{align*}\|P|_F-I_F\|<\eta. \end{align*}$$

Indeed, fix a normalised basis $\{e_1,\ldots ,e_k\}$ of $\operatorname {im} P$ , choose $x_{i_j}\in D$ sufficiently close to $e_j$ , and set $F=\operatorname {span}\{x_{i_j}:1\leqslant j\leqslant k\}$ ; for sufficiently close choices this space has dimension k and satisfies the displayed estimate. Applying Lemma 2.4 with $E=\operatorname {im}P$ , $T=P$ , and this F, we obtain a projection $P^\prime $ with $\operatorname {im} P^\prime \subseteq X_m$ . Since $\eta <\varepsilon $ , the estimates yielded by Lemma 2.4 imply the stated bounds.

Corollary 2.6. Let $ X $ be a Banach space with the $ {\mathscr {F}} $ - $ \pi $ -property. Suppose $ D = \{x_i \colon i \in \mathbb {N}\} $ is a dense subset of $ X $ . Define $ X_k $ as the linear span of $ \{x_1, \ldots , x_k\} $ for $ k \in \mathbb {N} $ . Then, there exists a sequence of projections $ (P_n^\prime )_{n=1}^\infty $ that witnesses the $ {\mathscr {F}} $ - $ \pi $ -property of $ X $ such that for every $ n \in \mathbb {N} $ , there exists $ m \in \mathbb {N} $ with $ \operatorname {im} P_n^\prime \subseteq X_m $ .

Proof. Fix any sequence of projections $ (P_n)_{n=1}^\infty $ witnessing the $ {\mathscr {F}} $ - $ \pi $ -property of $ X $ . Let $ (\varepsilon _n)_{n=1}^\infty $ be a sequence in $(0,1)$ such that

$$\begin{align*}\frac{\varepsilon_n \cdot \dim \operatorname{im} P_n \cdot \|P_n\|}{1 - \varepsilon_n} < \frac{1}{n}, \quad n \in \mathbb{N}. \end{align*}$$

For $ n \in \mathbb {N} $ , apply Corollary 2.5 to $ P_n $ and $ \varepsilon _n $ to obtain $ P_n^\prime $ . Note that for $ x \in X $ ,

$$\begin{align*}\|P_n^\prime x - x\| \leqslant \|P_n^\prime x - P_n x\| + \|P_n x - x\|, \end{align*}$$

and

$$\begin{align*}\|P_n^\prime x - P_n x\| \leqslant \frac{\varepsilon_n \cdot \dim \operatorname{im} P_n \cdot \|P_n\|}{1 - \varepsilon_n} \|x\|, \end{align*}$$

$$\begin{align*}\|P_n^\prime x - x\| \leqslant \frac{\varepsilon_n \cdot \dim \operatorname{im} P_n \cdot \|P_n\|}{1 - \varepsilon_n} \|x\| + \|P_n x - x\|. \end{align*}$$

Substituting the condition on $ \varepsilon _n $ , we have

$$\begin{align*}\|P_n^\prime x - x\| \leqslant \frac{1}{n} \|x\| + \|P_n x - x\|. \end{align*}$$

Hence, if $ P_n(x) \underset {n, {\mathscr {F}}}{\longrightarrow } x $ , it follows that $ P_n^\prime x \underset {n, {\mathscr {F}}}{\longrightarrow } x $ .

Moreover, we require a refinement of [Reference Casazza and Kalton3, Proposition 3.6].

Lemma 2.7. Let $ X $ be a Banach space, $\varepsilon>0$ , and let $ T \colon X \to X $ be a linear operator. There exists $0<\theta < \frac {1}{4}$ , such that if $ \|T - T^2\|< \theta $ , then there exists a projection $ P \colon X \to X $ such that

i) $\{x \in X \colon Tx = x\} \subseteq P[X] \subseteq T[X] $ ;
ii) $\|Px-x\|<\varepsilon $ whenever $\|Tx-x\|<\frac {\varepsilon }{2}$ and $x\in B_X$ ;
iii)
$$\begin{align*}\|P\| \leqslant \frac{1}{2} \left(1 + \frac{1 + 2\|T\|}{\sqrt{1 - 4\theta}} \right); \end{align*}$$
iv)
$$\begin{align*}\|P-T\|\leqslant \frac{1}{2}(2\|T\|+1)\cdot \left(\frac{1}{\sqrt{1-4\theta}} - 1 \right). \end{align*}$$

Moreover, $\theta $ may depend only on $\varepsilon $ and upper bound $\lambda \geqslant \|T\|$ , regardless of T, and such a dependence may be arranged to be continuous.

For our purposes, we require a slightly stronger statement, which can be derived using the same proof. Thus, we provide the full reasoning here for the sake of completeness. It seems to us that this stronger statement was already used in [Reference Casazza and Kalton3, Theorem 3.7] and [Reference Ghawadrah8], since in either case it is not mentioned why corrected operators have non-trivial fixed points.

Proof. Define

$$\begin{align*}S = \sum_{m=0}^\infty \binom{2m}{m} (T - T^2)^m\quad\text{ and }\quad P = \frac{1}{2}\big(I - (I - 2T)S\big). \end{align*}$$ upper S equals sigma summation Underscript m equals 0 Overscript infinity Endscripts StartBinomialOrMatrix 2 m Choose m EndBinomialOrMatrix left parenthesis upper T minus upper T squared right parenthesis Superscript m Baseline and upper P equals one half left parenthesis upper I minus left parenthesis upper I minus 2 upper T right parenthesis upper S right parenthesis period

The function $(1 - 4z)^{-\frac {1}{2}}$ admits a power series expansion given by

(2) $$ \begin{align} \sum_{m=0}^\infty \binom{2m}{m} z^m \quad\left(|z| < \frac{1}{4}\right). \end{align} $$ sigma summation Underscript m equals 0 Overscript infinity Endscripts StartBinomialOrMatrix 2 m Choose m EndBinomialOrMatrix z Superscript m Baseline left parenthesis StartAbsoluteValue z EndAbsoluteValue less than one fourth right parenthesis period

From this, it follows that $\|S\| \leqslant (1 - 4\theta )^{-\frac {1}{2}}$ . Moreover, observe that

$$\begin{align*}(I - 2T)^2 \;=\; I - 4\,T + 4\,T^2 \;=\; I \;-\; 4\,\bigl(T - T^2\bigr). \end{align*}$$

On the other hand, by definition of S we have, in the holomorphic functional-calculus sense, $ S \;=\; \bigl (1 - 4\,(T - T^2)\bigr )^{-\tfrac 12}. $ Hence $ S^2 \;=\; \bigl (1 - 4\,(T - T^2)\bigr )^{-1}. $ It follows directly that

$$\begin{align*}(I - 2T)^2 \, S^2 \;=\; \bigl[I - 4\,(T - T^2)\bigr] \;\cdot\; \bigl[1 - 4\,(T - T^2)\bigr]^{-1} \;=\; I. \end{align*}$$

Consequently, P is a projection on X, and the stated estimate for $\|P\|$ follows. Indeed,

$$\begin{align*}P^2 \;=\; \Bigl(\tfrac12\bigl[I - (I - 2T)\,S\bigr]\Bigr)^2 \;=\; \tfrac14\,\Bigl[I - (I - 2T)S \Bigr]^2. \end{align*}$$

$$\begin{align*}\bigl[I - (I - 2T)\,S\bigr]^2 \;=\; I \;-\; 2\,(I - 2T)\,S \;+\; (I - 2T)^2\,S^2, \end{align*}$$

we have

$$\begin{align*}P^2 \;=\; \tfrac14\,\bigl[\, I \;-\; 2\,(I - 2T)\,S \;+\; (I - 2T)^2\,S^2 \bigr], \end{align*}$$

$$\begin{align*}P^2 \;=\; \tfrac14\,\bigl[I + I - 2(I - 2T)\,S\bigr] \;=\; \tfrac12 \bigl[I - (I - 2T)\,S\bigr] \;=\; P. \end{align*}$$

Additionally, note that P can be expressed as

$$\begin{align*} P &= 3T^2 - 2T^3 + \frac{1}{2}(2T - I) \sum_{m=2}^\infty \binom{2m}{m} (T - T^2)^m \\ &= T + \frac{1}{2}(2T-I)\sum_{m=1}^\infty \binom{2m}{m} (T - T^2)^m. \end{align*}$$

StartLayout 1st Row 1st Column upper P 2nd Column equals 3 upper T squared minus 2 upper T cubed plus one half left parenthesis 2 upper T minus upper I right parenthesis sigma summation Underscript m equals 2 Overscript infinity Endscripts StartBinomialOrMatrix 2 m Choose m EndBinomialOrMatrix left parenthesis upper T minus upper T squared right parenthesis Superscript m Baseline 2nd Row 1st Column Blank 2nd Column equals upper T plus one half left parenthesis 2 upper T minus upper I right parenthesis sigma summation Underscript m equals 1 Overscript infinity Endscripts StartBinomialOrMatrix 2 m Choose m EndBinomialOrMatrix left parenthesis upper T minus upper T squared right parenthesis Superscript m Baseline period EndLayout

The remaining properties of P can be derived straightforwardly; in particular, $\theta $ may be chosen by the last equality. Indeed, the condition $ \|Px-x\|\leqslant \varepsilon $ follows from

$$\begin{align*}\frac{1}{2}(1+2\|T\|)\sum_{m=1}^\infty \binom{2m}{m}\theta^m<\frac{\varepsilon}{2},\quad \|x\|\leqslant 1,\quad \|Tx - x\|<\frac\varepsilon2. \end{align*}$$

Finally, as $P-T=\frac {1}{2}(2T-I)\sum _{m=1}^\infty \binom {2m}{m} (T - T^2)^m$ by (2), estimate iv follows.

Let us denote by $\Theta \colon (\lambda ,\varepsilon )\mapsto \theta $ the continuous map provided by the above lemma.

Corollary 2.8. Let X be a Banach space, ${\mathscr {F}}$ be a filter on $\mathbb {N}$ , and let $\Theta _n \colon \lambda \mapsto \Theta (\lambda , \frac 1n)$ . Suppose $(\lambda _n)_{n=1}^\infty $ is a sequence of real numbers all greater than $1$ .

Assume:

(1) $D := \{x_i : i \in \mathbb {N}\}$ is a dense subset of X.
(2) $(T_n)_{n=1}^\infty $ is a sequence of finite-rank operators on X such that
$$\begin{align*}\|T_n\|\;\leqslant\;\lambda_n \quad\text{and}\quad \|T_n - T_n^2\|\;\leqslant\;\Theta_n(\lambda_n) \quad\text{for all }n. \end{align*}$$
(3) For every $x \in B_X$ and every $\varepsilon>0$ , there is a set $A \in {\mathscr {F}}$ such that for each $n\in A$ one can find $i\in \mathbb {N}$ with
(3) $$ \begin{align} \|x - x_i\|\;\leqslant\; \frac{\varepsilon}{12\,\lambda_n} \quad\text{and}\quad \|x_i - T_n x_i\|\;\leqslant\;\frac\varepsilon4. \end{align} $$

Then X has the ${\mathscr {F}}$ - $\pi $ property.

Proof. Let $(P_n)_{n=1}^\infty $ be the sequence of improved projections associated with $(T_n)_{n=1}^\infty $ , provided by Lemma 2.7. We shall prove that $(P_n)_{n=1}^\infty $ witnesses the ${\mathscr {F}}$ - $\pi $ -property of X. Let us denote $ \theta _n=\Theta _n(\lambda _n)$ . By construction, each $P_n$ is a finite-rank projection in X. It remains to verify that for every $x \in X$ , the sequence $(P_n x)$ is ${\mathscr {F}}$ -convergent to x. Since ${\mathscr {F}}$ -convergence is determined on bounded sets, it suffices to establish this for $x \in B_X$ .

Fix $x \in B_X$ and $\varepsilon> 0$ . Let $A \in {\mathscr {F}}$ be given by (3). As ${\mathscr {F}}$ contains the Fréchet filter, we may assume that $\frac {1}{n} < \frac {\varepsilon }{2}$ for all $n \in A$ . For each $n \in A$ , there exists $i \in \mathbb {N}$ such that

$$\begin{align*}\|x - x_i\| \leqslant \frac{\varepsilon}{12 \lambda_n} \quad \text{and} \quad \|x_i - P_n x_i\| \leqslant \frac{\varepsilon}{2}. \end{align*}$$

Consequently, for every $n \in A$ , we estimate

$$\begin{align*}\|x - P_n x\| \leqslant \|x - x_i\| + \|x_i - P_n x_i\| + \|P_n x_i - P_n x\| \leqslant (1 + \|P_n\|) \|x - x_i\| + \frac{\varepsilon}{2}. \end{align*}$$

By the properties of $P_n$ , we further bound

$$\begin{align*}\|x - P_n x\| \leqslant \left(1 + \frac{1}{2} \cdot \frac{1 + 2\|T_n\|}{\sqrt{1 - 4\theta_n}} \right) \frac{\varepsilon}{12 \lambda_n} + \frac{\varepsilon}{2}. \end{align*}$$

Using the estimate $\|T_n\| \leqslant \lambda _n$ and $\theta _n \leqslant \frac {4}{5}$ , we obtain

$$\begin{align*}\|x - P_n x\| \leqslant \left( \lambda_n + \frac{1}{2} \cdot \frac{\lambda_n + 2\lambda_n}{\sqrt{1 - \frac{4}{5}}} \right) \frac{\varepsilon}{12 \lambda_n} + \frac{\varepsilon}{2} \leqslant \frac{6\varepsilon}{12} + \frac{\varepsilon}{2} = \varepsilon. \end{align*}$$

StartMetric x minus upper P Subscript n Baseline x EndMetric less than or slanted equals left parenthesis lamda Subscript n Baseline plus one half dot StartFraction lamda Subscript n Baseline plus 2 lamda Subscript n Baseline Over StartRoot 1 minus four fifths EndRoot EndFraction right parenthesis StartFraction epsilon Over 12 lamda Subscript n Baseline EndFraction plus StartFraction epsilon Over 2 EndFraction less than or slanted equals StartFraction 6 epsilon Over 12 EndFraction plus StartFraction epsilon Over 2 EndFraction equals epsilon period

This completes the proof.

3 Proof of Theorem A

In [Reference Ghawadrah7, Reference Ghawadrah8, Reference Godefroy and Saint-Raymond9], Ghawadrah dealt with the usual convergence, which allowed her to provide several properties under the same conditions. However, as we are investigating just ${\mathscr {F}}$ convergence rather than the usual one, we need to separate providing the existence of projections from providing their approximation properties. To be more precise, let us fix the linearly independent countable dense sequence $(x_n)$ in the separable Banach space X and denote $X_k=\operatorname {span}\{x_1, \ldots , x_k \}$ . Before formulating the main lemma, we define

$$\begin{align*}S:= \left\{(\sigma_{n}) \in (\mathbb{R}^{\mathbb{N}^2})^{\mathbb{N}}\colon \forall_{n} \exists_{m} \forall_{j \geqslant m} \forall_{i} \sigma_{n}(i,j) = 0 \right\}, \end{align*}$$

which is a $\Pi ^0_3$ -subset (hence Borel) of the product space $(\mathbb {R}^{\mathbb {N}^2})^{\mathbb {N}}$ .

Let us explain that the proper way of thinking of S is that n plays the role of picking $P_{n}$ and $i,j$ says we are considering the coefficient of the value of the constructed function in $x_i$ in the $x_j$ axis. Note that by [Reference Kechris13, Theorem 13.1], S may be viewed as a Polish space, so even though it appears rather complicated, quantifying over it is not a problem.

Recall also that ${\mathscr {F}}$ -projections, even if continuous, need not be uniformly bounded; therefore, we require the existence of a sequence $(\lambda _n)_{n=1}^\infty $ of constants responsible for the continuity of respective projections.

By $c_{00}^{\mathbb {Q}}$ we denote $\{(\alpha _n)\in c_{00}: \forall _n \alpha _n \in \mathbb {Q}\}$ . Let $\Theta _n\colon \lambda \mapsto \theta $ be as in Corollary 2.8.

Lemma 3.1. Let X be a Banach space. Then X has the ${\mathscr {F}}$ - $\pi $ -property if and only if the following condition holds:

$$ \begin{align*} \exists_{(\lambda_n)_{n=1}^\infty\in\mathbb{N}^{\mathbb{N}}} \ \exists{(\sigma_{n})\in S} \end{align*} $$

(4) $$ \begin{align} \forall_{(\alpha_i)\in c_{00}^{\mathbb{Q}}} \forall_{n\in \mathbb{N}} \left\| \sum_{i=1}^\infty \alpha_i \sum_{j=1}^\infty \sigma_{n}(i,j) x_j \right\| \leqslant \lambda_n \left\| \sum_{i=1}^\infty \alpha_i x_i \right\| \end{align} $$

for all Underscript left parenthesis alpha Subscript i Baseline right parenthesis element of c 00 Superscript double struck upper Q Baseline Endscripts for all Underscript n element of double struck upper N Endscripts StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis x Subscript j Baseline EndMetric less than or slanted equals lamda Subscript n Baseline StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline x Subscript i Baseline EndMetric

(5) $$ \begin{align} \forall_{(\alpha_i)\in c_{00}^{\mathbb{Q}}}\forall_{n\in\mathbb{N}} & \left\| \sum_{i=1}^\infty \alpha_i \sum_{j=1}^\infty \sigma_{n}(i,j) x_j - \sum_{i=1}^\infty \alpha_i\sum_{j=1}^\infty\sum_{t=1}^\infty \sigma_{n}(i,j)\sigma_{n}(j,t) x_t \right\| \nonumber \\ &\leqslant \Theta_n(\lambda_n) \left\|\sum_{i=1}^\infty \alpha_i x_i \right\| \end{align} $$

StartLayout 1st Row 1st Column for all Underscript left parenthesis alpha Subscript i Baseline right parenthesis element of c 00 Superscript double struck upper Q Endscripts for all Underscript n element of double struck upper N Endscripts 2nd Column StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis x Subscript j Baseline minus sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma summation Underscript t equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis sigma Subscript n Baseline left parenthesis j comma t right parenthesis x Subscript t Baseline EndMetric 2nd Row 1st Column Blank 2nd Column less than or slanted equals normal upper Theta Subscript n Baseline left parenthesis lamda Subscript n Baseline right parenthesis StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline x Subscript i Baseline EndMetric EndLayout

(6) $$ \begin{align} \forall_{x\in B_X} \forall_{r\in \mathbb{N}} \exists_{A\in{\mathscr{F}}} \forall_{n\in A}\exists_{i\in \mathbb{N}} \| x-x_i\| \leqslant \frac1r\cdot\frac{1}{12\lambda_n} \wedge \left\|x_i - \sum_{j=1}^\infty \sigma_{n}(i,j) x_j\right\| < \frac{1}{4r}. \end{align} $$

for all Underscript x element of upper B Subscript upper X Baseline Endscripts for all Underscript r element of double struck upper N Endscripts there exists Underscript upper A element of script upper F Endscripts for all Underscript n element of upper A Endscripts there exists Underscript i element of double struck upper N Endscripts StartMetric x minus x Subscript i Baseline EndMetric less than or slanted equals StartFraction 1 Over r EndFraction dot StartFraction 1 Over 12 lamda Subscript n Baseline EndFraction and StartMetric x Subscript i Baseline minus sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis x Subscript j Baseline EndMetric less than StartFraction 1 Over 4 r EndFraction period

Proof. “ $\Rightarrow $ ” Assume that $(P_n)_{n=1}^\infty $ is an ${\mathscr {F}}$ - $\pi $ -base for X. By Corollary 2.6, we may assume that $\operatorname {im}(P_n) \subset X_m$ for some $m \in \mathbb {N}$ . Set $\lambda _n :=\big \lceil \|P_n\| \big \rceil + 1$ and define $\mathbb {\sigma } \in S$ by the condition

$$\begin{align*}P_nx_i = \sum_{j=1}^\infty \sigma_{n}(i,j) x_j. \end{align*}$$ upper P Subscript n Baseline x Subscript i Baseline equals sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis x Subscript j Baseline period

The inclusion $\operatorname {im}(P_n) \subset X_m$ ensures that the above sum is in fact finite.

Verification of conditions (4) and 5 is straightforward (note that since $P_n$ ’s are projections, condition 5 is even simpler). To verify condition (6) let us fix $x\in B_X$ and $r\in \mathbb {N}$ . Since $(P_n)$ witnesses the ${\mathscr {F}}$ - $\pi $ -property of X, there is $A\in {\mathscr {F}}$ such that for every $n\in A \|P_nx-x\|\leqslant \frac {1}{8r}$ . Hence for $n \in A$ we may find $i\in \mathbb {N}$ such that

$$\begin{align*}\|x-x_i\|\leqslant \min\left\{\frac{1}{8r(1+\lambda_n)},\frac{1}{12r\lambda_n}\right\}, \end{align*}$$ StartMetric x minus x Subscript i Baseline EndMetric less than or slanted equals min left brace StartFraction 1 Over 8 r left parenthesis 1 plus lamda Subscript n Baseline right parenthesis EndFraction comma StartFraction 1 Over 12 r lamda Subscript n Baseline EndFraction right brace comma

and thus

$$\begin{align*}\|x_i-P_nx_i\| &\leqslant \|x-x_i\|+\|x-P_nx\|+ \|P_nx-P_n x_i\| \leqslant (1+\lambda_n)\|x-x_i\|+ \frac{1}{8r} \\ & \leqslant \frac{1}{4r}. \end{align*}$$

StartLayout 1st Row 1st Column StartMetric x Subscript i Baseline minus upper P Subscript n Baseline x Subscript i Baseline EndMetric 2nd Column less than or slanted equals StartMetric x minus x Subscript i Baseline EndMetric plus StartMetric x minus upper P Subscript n Baseline x EndMetric plus StartMetric upper P Subscript n Baseline x minus upper P Subscript n Baseline x Subscript i Baseline EndMetric less than or slanted equals left parenthesis 1 plus lamda Subscript n Baseline right parenthesis StartMetric x minus x Subscript i Baseline EndMetric plus StartFraction 1 Over 8 r EndFraction 2nd Row 1st Column Blank 2nd Column less than or slanted equals StartFraction 1 Over 4 r EndFraction period EndLayout

“ $\Leftarrow $ ” Suppose the proper $(\lambda _n)$ and $\sigma $ ’s are provided by our formula. The first step in the construction is to define

(7) $$ \begin{align} y_{n}^i = \sum_{j=1}^\infty \sigma_{n}(i, j) x_j, \end{align} $$ y Subscript n Superscript i Baseline equals sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis x Subscript j Baseline comma

and

(8) $$ \begin{align} z_{n}^i = \sum_{j=1}^\infty \sigma_{n}(i, j) y^i_{n} = \sum_{j=1}^\infty \sum_{t=1}^\infty \sigma_{n}(i, j) \sigma_{n}(j, t) x_t. \end{align} $$

By the definition of the set S, the above sums are finite. Let m denote their length. Since the sequence $(x_N)$ is linearly independent and dense in X, we can define operators $T_n \colon X \to X_m$ by the rule $T_n(x_i) = y_{n}^i$ . By condition (4), it follows that $\|T_n\| \leqslant \lambda _n$ , and by condition (5), the estimate $\|T_n - T_n^2\| \leqslant \Theta _n(\lambda _n)$ holds, since $T_n^2x_i=z^i_n$ .

Moreover, by (6) for every $x \in B_X$ and every $\varepsilon>0$ , there is a set $A \in {\mathscr {F}}$ such that for each $n\in A$ one can find $i\in \mathbb {N}$ with

(9) $$ \begin{align} \|x - x_i\|\;\leqslant\; \frac{\varepsilon}{12\,\lambda_n} \quad\text{and}\quad \|x_i - T_n x_i\|\;\leqslant\;\frac{\varepsilon}{4}. \end{align} $$

Hence we may apply Corollary 2.8 to conclude that X satisfies the ${\mathscr {F}}$ - $\pi $ -property.

We are ready to prove Theorem A.

Proof. Let $\{f_i:i\in \mathbb {N}\}$ be provided by Corollary 2.2 and set

$$\begin{align*}\mathbb{X}= \mathsf{SB} \times \mathbb{N}^{\mathbb{N}} \times S. \end{align*}$$

Moreover, let

$$ \begin{align*} \mathbb{X}_1 &= \bigcap_{(\alpha_i) \in c_{00}^{\mathbb{Q}}} \bigcap_{n \in \mathbb{N}} \\&\quad \Bigg\{ (X, (\lambda_n), \sigma)\in\mathbb{X}\colon\ \left\| \sum_{i=1}^\infty \alpha_i \sum_{j=1}^\infty \sigma_{n}(i, j) f_j(X) \right\| \leqslant \lambda_n \left\| \sum_{i=1}^\infty \alpha_i f_i(X) \right\| \Bigg\} \\\mathbb{X}_{2} &= \bigcap_{(\alpha_i) \in c_{00}^{\mathbb{Q}}}\bigcap_{n\in\mathbb{N}} \Bigg\{\left( X, (\lambda_n), \sigma \right)\in \mathbb{X} \colon \\&\quad \left\| \sum_{i=1}^\infty \alpha_i \sum_{j=1}^\infty \sigma_{n}(i, j) f_j(X) - \sum_{i=1}^\infty \alpha_i \sum_{j=1}^\infty \sum_{t=1}^\infty \sigma_{n}(i, j) \sigma_{n}(j, t) f_t(X) \right\| \\ &\quad \leqslant \Theta_n(\lambda_n) \left\| \sum_{i=1}^\infty \alpha_i f_i(X) \right\| \Bigg\} \end{align*} $$

StartLayout 1st Row 1st Column double struck upper X 1 2nd Column equals intersection Underscript left parenthesis alpha Subscript i Baseline right parenthesis element of c 00 Superscript double struck upper Q Baseline Endscripts intersection Underscript n element of double struck upper N Endscripts 2nd Row 1st Column Blank 2nd Column StartSet left parenthesis upper X comma left parenthesis lamda Subscript n Baseline right parenthesis comma sigma right parenthesis element of double struck upper X colon StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis f Subscript j Baseline left parenthesis upper X right parenthesis EndMetric less than or slanted equals lamda Subscript n Baseline StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline f Subscript i Baseline left parenthesis upper X right parenthesis EndMetric EndSet 3rd Row 1st Column double struck upper X 2 2nd Column equals intersection Underscript left parenthesis alpha Subscript i Baseline right parenthesis element of c 00 Superscript double struck upper Q Baseline Endscripts intersection Underscript n element of double struck upper N Endscripts left brace left parenthesis upper X comma left parenthesis lamda Subscript n Baseline right parenthesis comma sigma right parenthesis element of double struck upper X colon 4th Row 1st Column Blank 2nd Column StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis f Subscript j Baseline left parenthesis upper X right parenthesis minus sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma summation Underscript t equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis sigma Subscript n Baseline left parenthesis j comma t right parenthesis f Subscript t Baseline left parenthesis upper X right parenthesis EndMetric 5th Row 1st Column Blank 2nd Column less than or slanted equals normal upper Theta Subscript n Baseline left parenthesis lamda Subscript n Baseline right parenthesis StartMetric sigma summation Underscript i equals 1 Overscript infinity Endscripts alpha Subscript i Baseline f Subscript i Baseline left parenthesis upper X right parenthesis EndMetric right brace EndLayout

and for $r\in \mathbb {N}$ set

$$ \begin{align*} \mathbb{W}_{r} &= \bigcap_{n\in\mathbb{N}} \bigcup_{i \in \mathbb{N}} \Bigg\{ (X, (\lambda_n), \sigma, x, A)\in \mathbb{X}\times B_{C(\Delta)}\times \{0,1\}^{\mathbb{N}} \colon \\ &\quad n \notin A \vee \left( \| x-f_i(X)\| \leqslant \frac1r\cdot\frac{1}{12\lambda_n} \wedge \left\|f_i(X) - \sum_{j=1}^\infty \sigma_{n}(i,j) f_j(X)\right\| < \frac{1}{4r}\right) \Bigg\}. \end{align*} $$

StartLayout 1st Row 1st Column double struck upper W Subscript r 2nd Column equals intersection Underscript n element of double struck upper N Endscripts union Underscript i element of double struck upper N Endscripts left brace left parenthesis upper X comma left parenthesis lamda Subscript n Baseline right parenthesis comma sigma comma x comma upper A right parenthesis element of double struck upper X times upper B Subscript upper C left parenthesis normal upper Delta right parenthesis Baseline times left brace 0 comma 1 right brace Superscript double struck upper N Baseline colon 2nd Row 1st Column Blank 2nd Column n not an element of upper A or left parenthesis StartMetric x minus f Subscript i Baseline left parenthesis upper X right parenthesis EndMetric less than or slanted equals StartFraction 1 Over r EndFraction dot StartFraction 1 Over 12 lamda Subscript n Baseline EndFraction and StartMetric f Subscript i Baseline left parenthesis upper X right parenthesis minus sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis f Subscript j Baseline left parenthesis upper X right parenthesis EndMetric less than StartFraction 1 Over 4 r EndFraction right parenthesis right brace period EndLayout

Note that if $(X, (\lambda _n), \sigma , x, A)\in \mathbb {W}_r$ , then $x\in X$ .

Given a product space $A\times B$ , by $\Pi _A\colon A\times B \to A$ we mean the projection onto the A-coordinate. Let

$$\begin{align*}\mathbb{V}_r=\mathbb{W}_r \cap \mathbb{X}\times B_{C(\Delta)}\times {\mathscr{F}}, \ \mathbb{V}=\bigcap_{r\in\mathbb{N}} \Pi_{\mathbb{X}\times B_{C(\Delta)}} [\mathbb{V}_r]. \end{align*}$$

Finally set

$$\begin{align*}\mathbb{X}_3 = \mathbb{X} \setminus \Pi_{\mathbb{X}}\left[\left( \mathbb{X} \times B_{C(\Delta)}\right)\setminus \mathbb{V}\right]. \end{align*}$$

Note that by Lemma 3.1 $\mathsf {SB}_{{\mathscr {F}}-\pi }=\Pi _{\mathsf {SB}}\left [\mathbb {X}_{1}\cap \mathbb {X}_{2} \cap \mathbb {X}_{3}\right ]$ . Moreover, if ${\mathscr {F}}$ is $\Sigma _s^1$ , then clearly $\mathbb {X}_{1},\mathbb {X}_{2}$ , and $\mathbb {W}_{r}$ are Borel, $\mathbb {V}_{r}, \mathbb {V}$ are $\Sigma _s^1$ , thus $\mathbb {X}_3$ is $\Pi ^1_{s+1}$ and we conclude that $\mathsf {SB}_{{\mathscr {F}}-\pi }$ is $\Sigma _{s+2}^1$ .

In order to get the claim concerning countably generated filter ${\mathscr {F}}$ generated by sets $\{A_l\colon l\in \mathbb {N}\}$ , one needs to provide analogous reasoning, yet in the condition (6) of Lemma 3.1 projective quantifier $\exists _{A\in {\mathscr {F}}}$ may be replaced by the countable one $\exists _{l\in \mathbb {N}}$ with further referring to $A_l$ instead of A. In such a case we set

$$ \begin{align*} \mathbb{V'} &= \bigcap_{r\in\mathbb{N}} \bigcup_{l\in\mathbb{N}} \bigcap_{n\in\mathbb{N}} \bigcup_{i \in \mathbb{N}} \Bigg\{ (X, (\lambda_n), \sigma, x)\in \mathbb{X}\times B_{C(\Delta)}\times \{0,1\}^{\mathbb{N}} \colon \\ &\quad n \notin A_l \vee \left( \| x-f_i(X)\| \leqslant \frac1r\cdot\frac{1}{12\lambda_n} \wedge \left\|f_i(X) - \sum_{j=1}^\infty \sigma_{n}(i,j) f_j(X)\right\| < \frac{1}{4r}\right) \Bigg\} \end{align*} $$

StartLayout 1st Row 1st Column double struck upper V prime 2nd Column equals intersection Underscript r element of double struck upper N Endscripts union Underscript l element of double struck upper N Endscripts intersection Underscript n element of double struck upper N Endscripts union Underscript i element of double struck upper N Endscripts left brace left parenthesis upper X comma left parenthesis lamda Subscript n Baseline right parenthesis comma sigma comma x right parenthesis element of double struck upper X times upper B Subscript upper C left parenthesis normal upper Delta right parenthesis Baseline times left brace 0 comma 1 right brace Superscript double struck upper N Baseline colon 2nd Row 1st Column Blank 2nd Column n not an element of upper A Subscript l Baseline or left parenthesis StartMetric x minus f Subscript i Baseline left parenthesis upper X right parenthesis EndMetric less than or slanted equals StartFraction 1 Over r EndFraction dot StartFraction 1 Over 12 lamda Subscript n Baseline EndFraction and StartMetric f Subscript i Baseline left parenthesis upper X right parenthesis minus sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis f Subscript j Baseline left parenthesis upper X right parenthesis EndMetric less than StartFraction 1 Over 4 r EndFraction right parenthesis right brace EndLayout

along with

$$\begin{align*}\mathbb{X'}_3 = \mathbb{X} \setminus \Pi_{\mathbb{X}}\left[\left( \mathbb{X} \times B_{C(\Delta)}\right)\setminus \mathbb{V'}\right]. \end{align*}$$

In such a case sets $\mathbb {X}_{1},\mathbb {X}_{2}$ , and $\mathbb {V'}$ are Borel, yet $\mathbb {X'}_{3}$ is $\Pi _1^1$ , so $\mathsf {SB}_{{\mathscr {F}}-\pi }$ is $\Sigma ^1_2$ .

Although we were unable to reduce the problem to a genuine $\Sigma ^1_3$ set, we conjecture that the formula above is not substantially simpler. Achieving such a reduction would likely require deriving convergence on the entire space X from convergence on a dense subset, or encoding ${\mathscr {F}}$ -convergence using countable quantifiers. A significant obstacle to this approach is the potential failure of uniform boundedness of the norms of the witnessing projections. This consideration also motivates the formulation of our condition, which begins with the existence of the sequence $(\lambda _n)_{n=1}^\infty $ . This existential quantification already places the complexity beyond the Borel hierarchy and obviates the need to enforce rationality of the $\sigma $ ’s. Indeed, even restricting quantification to a countable set S would not suffice to reduce the resulting complexity.

Remark 1. One might ask whether our result justifies the effort involved, as the problem might initially appear straightforward. A natural first attempt could be to express the property as the formula:

$$\begin{align*}\exists_{(P_n)} \forall_{x \in X} \forall_{\varepsilon> 0} \exists_{A \in {\mathscr{F}}} \forall_{n \in A} \|x - P_n x\| < \varepsilon, \end{align*}$$

which, at first glance, resembles the definition of a $\Sigma ^1_3$ set (possibly modulo some technical details). However, this is not the case. The initial existential quantifier requires the existence of a sequence of projections, but the space over which these projections are considered is non-trivial to define.

One might initially consider the space of all bounded linear operators $B(X, X)$ . However, this space is typically not separable in the norm topology, and its unit ball is Polish in the SOT only if $X^*$ is separable. A more refined approach would restrict attention to the space of finite-rank operators, which is separable (in either norm or SOT) only if $X^*$ is separable. Consequently, directly deducing the final projective complexity class from such a formula appears infeasible without employing the detailed approximation arguments presented earlier in this work.

4 Proof of Theorems B and C

This section is heavily inspired by the proof of [Reference de Rancourt, Kania and Swaczyna15, Theorem B].

Proof of Theorem B.

Set

$$\begin{align*}\mathcal{A}:= \Bigg\{A \subset \mathbb{N}\colon \exists_{x \in X} \exists_{\varepsilon>0}\; A \supset \Big\{ n \in \mathbb{N}\colon \big\|P_nx - x \big\| \leqslant \varepsilon \Big\}\Bigg\}. \end{align*}$$ script upper A colon equals StartSet upper A subset of double struck upper N colon there exists Underscript x element of upper X Endscripts there exists Underscript epsilon greater than 0 Endscripts upper A superset of StartSet n element of double struck upper N colon StartMetric upper P Subscript n Baseline x minus x EndMetric less than or slanted equals epsilon EndSet EndSet period

Clearly $\mathcal {A} \subset {\mathscr {F}}$ . Note that $(P_n)$ witnesses that X has ${\mathscr {F}}^\prime $ - $\pi $ -property for any filter ${\mathscr {F}}'$ on $\mathbb {N}$ such that $\mathcal {A} \subset {\mathscr {F}}' $ . Now, for every $n \in \mathbb {N}$ , consider the set

$$\begin{align*}\mathcal{B}_n= \Big\{ (x,\varepsilon,A)\in X \times (0,\infty)\times \{0, 1\}^{\mathbb{N}}\colon n \in A \vee \big\|P_nx - x \big\|> \varepsilon \Big\}. \end{align*}$$

By the continuity of $P_n$ we obtain that $\mathcal {B}_n$ is open. Observing that

$$\begin{align*}\mathcal{A} = \operatorname{proj}_{\{0, 1\}^{\mathbb{N}}}\Big[\bigcap_{n \in \mathbb{N}} \mathcal{B}_n\Big], \end{align*}$$ script upper A equals proj Subscript left brace 0 comma 1 right brace Sub Superscript double struck upper N Subscript Baseline left bracket intersection Underscript n element of double struck upper N Endscripts script upper B Subscript n Baseline right bracket comma

we deduce that $\mathcal {A}$ is analytic.

As $\mathcal {A} \subset {\mathscr {F}}$ , finite intersections of elements of $\mathcal {A}$ are readily non-empty. Consequently, $\mathcal {A}$ generates a filter ${\mathscr {F}}' \subset {\mathscr {F}}$ . We have

$$\begin{align*}\begin{aligned} {\mathscr{F}}^\prime &= \{A \subset \mathbb{N}\colon \exists_{n \in \mathbb{N}} \exists_{A_1, A_2, \ldots, A_n \in \mathcal{A}} A \supset A_1 \cap A_2 \cap \cdots \cap A_n \} \\&= \bigcup_{n \in \mathbb{N}} \operatorname{proj}_{(\{0, 1\}^{\mathbb{N}})_1} [\{ (A, A_1, \ldots, A_n)\in (\{0, 1\}^{\mathbb{N}})^{n+1}\colon \\& \quad A \supset A_1 \cap \cdots \cap A_n \wedge A_1, \ldots, A_n \in \mathcal{A}\}], \end{aligned} \end{align*}$$

StartLayout 1st Row 1st Column script upper F prime 2nd Column equals StartSet upper A subset of double struck upper N colon there exists Underscript n element of double struck upper N Endscripts there exists Underscript upper A 1 comma upper A 2 comma ellipsis comma upper A Subscript n Baseline element of script upper A Endscripts upper A superset of upper A 1 intersection upper A 2 intersection midline horizontal ellipsis intersection upper A Subscript n Baseline EndSet 2nd Row 1st Column Blank 2nd Column equals union Underscript n element of double struck upper N Endscripts proj Subscript left parenthesis left brace 0 comma 1 right brace Sub Superscript double struck upper N Subscript right parenthesis Sub Subscript 1 Subscript Baseline left bracket left brace left parenthesis upper A comma upper A 1 comma ellipsis comma upper A Subscript n Baseline right parenthesis element of left parenthesis left brace 0 comma 1 right brace Superscript double struck upper N Baseline right parenthesis Superscript n plus 1 Baseline colon 3rd Row 1st Column Blank 2nd Column upper A superset of upper A 1 intersection midline horizontal ellipsis intersection upper A Subscript n Baseline and upper A 1 comma ellipsis comma upper A Subscript n Baseline element of script upper A right brace right bracket comma EndLayout

where $\operatorname {proj}_{(\{0, 1\}^{\mathbb {N}})_1}$ is the projection onto the first coordinate. Consequently, ${\mathscr {F}}'$ is an analytic filter and $(P_n)_{n=1}^\infty $ witnesses that X has ${\mathscr {F}}'$ - $\pi $ -property since $\mathcal {A}\subset {\mathscr {F}}' \subset {\mathscr {F}}$ .

Proof of Theorem C.

We start with describing the family of all analytic filters on $\mathbb {N}$ . Recall that set B is analytic within Polish space Y if and only if there exists a closed subset $D\subset Y\times \mathbb {N}^{\mathbb {N}}$ such that $B=\Pi _Y[D]$ , while the space $F(Y\times \mathbb {N}^{\mathbb {N}})$ is a standard Borel space while equipped with the Effros–Borel structure. We may consider the following sets:

$$ \begin{align*} \mathbb{INT} &= \bigg\{D\in F\left(\{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}\right)\colon \forall_{(a_1,b_1),(a_2,b_2)\in \{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}} \exists_{(a_3,b_3)\in \{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}} \\& \qquad (a_1,b_1)\notin D \vee (a_2,b_2)\notin D \vee \left[ a_3=a_1\cap a_2 \wedge (a_3,b_3)\in D\right] \bigg\},\\\mathbb{SUP} &= \bigg\{D\in F\left(\{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}\right)\colon \forall_{(a_1,b_1)\in \{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}}\forall_{a\in\{0,1\}^{\mathbb{N}}} \exists_{(a_2,b_2)\in \{0,1\}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}} \\&\qquad (a_1,b_1)\notin D \vee a\not\supset a_1 \vee \left[a=a_2 \wedge (a_2,b_2) \in D\right] \bigg\}. \end{align*} $$

StartLayout 1st Row 1st Column double struck upper I upper N upper T 2nd Column equals left brace upper D element of upper F left parenthesis left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline right parenthesis colon for all Underscript left parenthesis a 1 comma b 1 right parenthesis comma left parenthesis a 2 comma b 2 right parenthesis element of left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline Endscripts there exists Underscript left parenthesis a 3 comma b 3 right parenthesis element of left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline Endscripts 2nd Row 1st Column Blank 2nd Column left parenthesis a 1 comma b 1 right parenthesis not an element of upper D or left parenthesis a 2 comma b 2 right parenthesis not an element of upper D or left bracket a 3 equals a 1 intersection a 2 and left parenthesis a 3 comma b 3 right parenthesis element of upper D right bracket right brace comma 3rd Row 1st Column double struck upper S upper U upper P 2nd Column equals left brace upper D element of upper F left parenthesis left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline right parenthesis colon for all Underscript left parenthesis a 1 comma b 1 right parenthesis element of left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline Endscripts for all Underscript a element of left brace 0 comma 1 right brace Superscript double struck upper N Baseline Endscripts there exists Underscript left parenthesis a 2 comma b 2 right parenthesis element of left brace 0 comma 1 right brace Superscript double struck upper N Baseline times double struck upper N Superscript double struck upper N Baseline Endscripts 4th Row 1st Column Blank 2nd Column left parenthesis a 1 comma b 1 right parenthesis not an element of upper D or a not a superset of a 1 or left bracket a equals a 2 and left parenthesis a 2 comma b 2 right parenthesis element of upper D right bracket right brace period EndLayout

Note that $\mathbb {INT}$ stands for family of those D whose projections are closed with respect to intersections of its elements, while $\mathbb {SUP}$ stands for family of those D whose projections are closed with respect to supersets of its elements. We may hence define

$$\begin{align*}\mathbb{AF}=\mathbb{SUP}\ \cap\ \mathbb{INT}\end{align*}$$

and observe that ${\mathscr {F}}\subset \{0,1\}^{\mathbb {N}}$ is an analytic filter if and only if there is a $D\in \mathbb {AF}$ such that ${\mathscr {F}}=\Pi _{\{0,1\}^{\mathbb {N}}}[D]$ .

Moreover, it is straightforward to verify that both $\mathbb {SUP},\mathbb {INT}$ are $\Pi ^1_2$ , since for any Polish Y and $y\in Y$ set ${\{D{\kern-1pt}\in{\kern-1pt} F(Y)\colon y\in D\}{\kern-1pt}={\kern-1pt}\bigcap _{n\in \mathbb {N}} \{D\in F(Y)\colon D\cap B(y,\frac 1n){\kern-1pt}\neq{\kern-1pt} \emptyset \}}$ is Borel within $F(Y)$ .

Let $\mathbb {X}, \mathbb {X}_1$ , and $\mathbb {X}_2$ be as in the proof of Theorem A, and set $ \mathbb {Y}=\mathbb {X}\times F\left (\{0,1\}^{\mathbb {N}}\times \mathbb {N}^{\mathbb {N}}\right ), \mathbb {Y}_1=\mathbb {X}_1\times F\left (\{0,1\}^{\mathbb {N}}\times \mathbb {N}^{\mathbb {N}}\right ), \mathbb {Y}_2=\mathbb {X}_2\times F\left (\{0,1\}^{\mathbb {N}}\times \mathbb {N}^{\mathbb {N}}\right )$ . Moreover, for $r\in \mathbb {N}$ set

$$ \begin{align*} & \mathbb{U}_{r} = \bigcap_{n\in\mathbb{N}} \bigcup_{i \in \mathbb{N}} \Bigg\{ (X, (\lambda_n), \sigma, D, x, A)\in \mathbb{Y}\times B_{C(\Delta)}\times \{0,1\}^{\mathbb{N}} \colon A \in \Pi_{\{0,1\}^{\mathbb{N}}}[D] \wedge \\&\qquad \left[\! n\notin A \vee \!\left(\! \| x-f_i(X)\| \leqslant \frac1r\cdot\frac{1}{12\lambda_n} \wedge \!\left\|f_i(X) - \sum_{j=1}^\infty \sigma_{n}(i,j) f_j(X)\right\|\! < \frac{1}{4r} \!\right)\!\right]\! \Bigg\}. \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column double struck upper U Subscript r Baseline equals intersection Underscript n element of double struck upper N Endscripts union Underscript i element of double struck upper N Endscripts left brace left parenthesis upper X comma left parenthesis lamda Subscript n Baseline right parenthesis comma sigma comma upper D comma x comma upper A right parenthesis element of double struck upper Y times upper B Subscript upper C left parenthesis normal upper Delta right parenthesis Baseline times left brace 0 comma 1 right brace Superscript double struck upper N Baseline colon upper A element of normal upper Pi Subscript left brace 0 comma 1 right brace Sub Superscript double struck upper N Subscript Baseline left bracket upper D right bracket and 2nd Row 1st Column Blank 2nd Column left bracket n not an element of upper A or left parenthesis StartMetric x minus f Subscript i Baseline left parenthesis upper X right parenthesis EndMetric less than or slanted equals StartFraction 1 Over r EndFraction dot StartFraction 1 Over 12 lamda Subscript n Baseline EndFraction and StartMetric f Subscript i Baseline left parenthesis upper X right parenthesis minus sigma summation Underscript j equals 1 Overscript infinity Endscripts sigma Subscript n Baseline left parenthesis i comma j right parenthesis f Subscript j Baseline left parenthesis upper X right parenthesis EndMetric less than StartFraction 1 Over 4 r EndFraction right parenthesis right bracket right brace period EndLayout

Note that

$$\begin{align*}A \in \Pi_{\{0,1\}^{\mathbb{N}}}[D] \Leftrightarrow \exists_{b\in\mathbb{N}^{\mathbb{N}}} (A,b)\in D, \end{align*}$$

so $\mathbb {U}_{r}$ is $\Sigma _1^1$ . Now let

$$\begin{align*}\mathbb{U}=\bigcap_{r\in\mathbb{N}} \Pi_{\mathbb{Y}\times B_{C(\Delta)}} [\mathbb{U}_r] \text{ and } \mathbb{Y}_3 = \mathbb{Y} \setminus \Pi_{\mathbb{Y}}\left[\left( \mathbb{Y} \times B_{C(\Delta)}\right)\setminus \mathbb{U}\right]. \end{align*}$$

Note that by Lemma 3.1 and Theorem B

$$\begin{align*}\mathsf{SB}_{\text{Filter-}\pi}=\Pi_{\mathsf{SB}}\left[\mathbb{Y}_{1}\cap \mathbb{Y}_{2} \cap \mathbb{Y}_{3}\cap\left(\mathbb{X}\times\mathbb{AF}\right)\right]. \end{align*}$$

Observe that $\mathbb {Y}_{1},\mathbb {Y}_{2}$ are Borel, $\mathbb {U}$ is $\Sigma _1^1$ , thus $\mathbb {Y}_3$ is $\Pi ^1_{2}$ . Set $\mathbb {X}\times \mathbb {AF}$ is also $\Pi ^1_2$ , so we conclude that $\mathsf {SB}_{\text {Filter-}\pi }$ is $\Sigma _{3}^1$ .

5 Problems

Question 5.1. In the analysis of the descriptive complexity of the set $\mathsf {SB}_{\mathrm {Fr}\text {-}\pi }$ by Ghawadrah [Reference Ghawadrah8], a choice of dense sequences in a separable Banach space is employed to characterise the $\pi $ -property with respect to the Fréchet filter. It remains unclear whether her approach presupposes the linear independence of this sequence; however this seems necessary. Furthermore, the claimed Borel complexity of $\Sigma ^0_6$ for $\mathsf {SB}_{\mathrm { Fr}\text {-}\pi }$ appears to hinge on the definition of certain operators $T_{K,\varepsilon }$ , whose well-posedness may be inadequately justified. Let us then state the following problem formally:

is there a continuous (with respect to an admissible topology) way of picking dense and linearly independent subsets of separable Banach spaces?

Funding

Funding received from NCN Project SONATA BIS 13 No. 2023/50/E/ST1/00067 is acknowledged with thanks. The first-named author acknowledges the Institute of Mathematics, Czech Academy of Sciences; RVO: 67985840.

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