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On a weak topology for Hadamard spaces A. B¸rd¸llima Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 10, DOI: https://doi.org/10.4213/sm9808 
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    § 1. IntroductionEvery metric space can be equipped with a canonical topology induced by its metric. This topology characterizes the usual (strong) convergence of sequences. It has been of interest to analysts to study weaker notions of convergence for sequences in a metric space and, if possible, to identify topologies that induce such convergences. In this work we consider the notion of weak convergence introduced by Jost [12] in the setting of a Hadamard space, and our aim is to identify the most natural topology associated with it. On bounded sets Jost’s notion, which we simply refer to as weak convergence, coincides with $\Delta$-convergence as defined earlier by Lim [16]. The topological characterization of weak convergence for bounded sequences was resolved recently by Lytchak and Petrunin in [17], who were answering a question posed by Bačak in [8], originally in the setting of $\Delta$-convergence. However the problem of constructing a topology that also characterizes weak convergence of unbounded sequences remained open. Dropping the requirement of boundedness in our study of weak convergence highlights one of the key differences between a general Hadamard space and Hilbert spaces, since for the latter the Banach-Steinhaus theorem implies that every weakly convergent sequence must be bounded. Inspired by a question of Bačak [8] concerning $\Delta$-convergence, the author proposed incidentally in his thesis [3] such a weak topology $\tau_w$ on what we call weakly proper Hadamard spaces. Weak properness entails a certain topological regularity condition for open sets of $\tau_w$. While a wide range of Hadamard spaces are weakly proper, among which are Hilbert spaces and all locally compact Hadamard spaces, not all of them enjoy this property as witnessed by the example in [17], § 4. The theory about $\tau_w$ can be accommodated in more generality for nets. Such an approach is taken in our subsequent work [19]. Our work is a presentation of the main results in [3], and it follows the original arguments there. We start in § 2 with some preliminaries from the general theory about Hadamard spaces. Then we construct the weak topology $\tau_w$ in § 3, and introduce the notion of a weakly proper Hadamard space. We show that on a weakly proper Hadamard space, weak sequential convergence and weak convergence coincide (Theorem 3.1) and, in particular, such a space is always Hausdorff in $\tau_w$ (Lemma 3.2). Moreover, many useful properties of the weak topology in a Hilbert space extend naturally to the setting of a Hadamard space. For example, a (geodesically) convex set is closed if and only if it is weakly closed (Theorem 3.2). In particular, we include some findings which do not appear in [19]. For instance, we show that a result of Mazur’s lemma type holds (Theorem 3.3). Furthermore, in the class of weakly proper separable Hadamard spaces, provided that the projections onto geodesics satisfy a certain regularity condition, weak compactness and weak sequential compactness coincide (Theorems 3.5 and 3.6). This is reminiscent of the Eberlein-Šmulyan Theorem (see [24] and [2], for example). It is unknown if all Hadamard spaces enjoy this regularity condition, and we leave this as an open problem. Another aspect we treat are locally compact Hadamard spaces; in particular, we show that a weakly convergent sequence (net) is always bounded (Lemma 3.5), independently of the result that weak and strong topology coincide in such spaces (Theorem 3.8). Another consequence is that every locally compact Hadamard space is weakly proper (Corollary 3.5). We then bring up the case of a Hilbert space and show that a Hilbert space is weakly proper and that $\tau_w$ coincides with the usual weak topology there (Proposition 3.5). Moreover, the product of a locally compact Hadamard space and a Hilbert space is always weakly proper (Corollary 3.7). Finally, in § 4 we compare our notion to some existing notions of weak topology, in particular, to the one due to Kakavandi (Theorem 4.1). We conclude in § 4.4 with a short discussion about the weak topology introduced recently by Lytchak and Petrunin [17]. § 2. PreliminariesThroughout, we let $(H,d)$ denote a Hadamard space. A constant-speed geodesic $\gamma\colon [0,1]\to H$ is a curve satisfying the condition $d(\gamma(s), \gamma(t))=|s-t|\,d(\gamma(0),\gamma(1))$ for all $s,t\in [0,1]$. Given $x\in H$, we define $\Gamma_{x}(H)$ to be the set of all constant speed geodesics $\gamma\colon [0,1]\to H$ such that $\gamma(0)=x$. Given $x,y\in H$, we denote by $[x,y]$ the geodesic segment joining $x$ and $y$ and by $x_t=(1-t)x\oplus t y$ the element in $[x,y]$ such that $d(x_t,x)=td(x,y)$ for $t\in[0,1]$. The geodesic triangle $\Delta(p, q, r)$ determined by the three points $p,q,r\in H$ consists of the three geodesic segments $[p,q]$, $[q,r]$ and $[r,p]$. To each geodesic triangle $\Delta(p, q, r)$ there corresponds a comparison triangle $\Delta(\overline{p}, \overline{q}, \overline{r})$ in the Euclidean plane $\mathbb{E}^2$ with vertices $\overline{p}$, $\overline{q}$ and $\overline{r}$ such that $d(p,q)=\|\overline{p}-\overline{q}\|$, $d(q,r)=\|\overline{q}-\overline{r}\|$ and $d(p,r)=\|\overline{p}-\overline{r}\|$. A point $\overline{x}\in[\overline p,\overline q]$ is a comparison point in $\mathbb E^2$ for the point $x\in[p,q]$ if $d(p,x)=\|\overline p-\overline x\|$. For every $x\in[p,q]$ and $y\in[p,r]$ a Hadamard space satisfies the $\operatorname{CAT}(0)$-inequality Equivalently a Hadamard space is called a complete $\operatorname{CAT}(0)$-space (see [5], [6] and [22], for example). The inequality (2.1) entails nonpositive curvature, which characterizes $\operatorname{CAT}(0)$-spaces. It is in terms of this inequality that metric spaces with a notion of bounded curvature were originally defined by Alexandrov in [1] and later popularized by Gromov in [10] and [11].Another concept we need is that of an angle. The internal angle of $\Delta(\overline{p}, \overline{q}, \overline{r})$ at $\overline{p}$ is called the comparison angle between $q$ and $r$ at $p$ and denoted by $\overline{\angle}_p(q,r)$. Now let $\gamma$ and $\eta$ be two geodesic segments emanating from the same point $p$, that is, $\gamma(0)=\eta(0)=p$. The Alexandrov angle between $\gamma$ and $\eta$ at $p$ is the number $\angle_p(\gamma,\eta)\in[0,\pi]$ defined by
$$
\begin{equation}
\angle_p(\gamma,\eta):=\limsup_{t,t'\to 0}\overline{\angle}_p(\gamma(t),\eta(t')).
\end{equation}
\tag{2.2}
$$
On a Riemannian manifold the Alexandrov angle coincides with the usual one. A set $C\subseteq H$ is (geodesically) convex if $x,y\in C$ implies that $[x,y]\subseteq C$. We denote by $\operatorname{co} C$ the convex hull of $C$, that is, the smallest convex set in $H$ that contains $C$ entirely. Let $d(x,C)=\inf\{y\in C\colon d(x,y)\}$, and denote by $P_Cx:=\{{y\in H}\colon d(x,y)=d(x,C)\}$ the metric projection of $x$ onto $C$. In general, $P_Cx$ can be multivalued or even empty; however, when $C$ is a closed and convex set then $P_Cx$ is nonempty and unique for every $x\in H$. Moreover, by virtue of Theorem 2.1.12 in [7] the following inequality is satisfied:
$$
\begin{equation}
d(y,P_Cx)^2+d(x,P_Cx)^2\leqslant d(x,y)^2 \quad\text{for all} \ y\in C.
\end{equation}
\tag{2.3}
$$
From inequality (2.1) another equivalent characterization of $\operatorname{CAT}(0)$-spaces, and therefore also of Hadamard spaces, follows:
$$
\begin{equation}
d(x_t,z)^2\leqslant(1-t)d(x,z)^2+td(y,z)^2-t(1-t)d(x,y)^2, \qquad x,y,z\in H,\quad t\in[0,1].
\end{equation}
\tag{2.4}
$$
We conclude this section by the central notion of weak convergence as given by Jost in [12]. Definition 2.1. Let $(x_n)\subseteq H$ and $x\in H$. We say that $x_n$ converges weakly to $x$ and we denote this by $x_n\xrightarrow{w} x$ if $\lim_{n\to\infty}P_{\gamma}x_n=x$ for every $\gamma\in\Gamma_x(H)$. Note that this definition extends to nets in a straightforward manner. We also point out that a similar notion appears to have later been studied independently by Sosov in [23]. Remark 2.1. Weak limits are unique. Indeed if $x_n\xrightarrow{w} x$ and $x_n\xrightarrow{w} y$, then $0\leqslant d(x,y)\leqslant d(P_{[x,y]}x_n,x)+d(P_{[x,y]}x_n,y)\to 0$ as $n\to\infty$ implies that $x=y$. § 3. Identification of a weak topology3.1. The construction of open setsWe follow the standard terminology in topology (see [18], [15] and [21], for example). Consider a Hadamard space $(H,d)$. Following [8] we say that a set $U\subset H$ is weakly open if for every $x\in U$ there exists some $\varepsilon>0$ and a finite family of geodesics $\gamma_1,\gamma_2,\dots,\gamma_n \in \Gamma_{x}(H)$ such that the set
$$
\begin{equation}
U_{x}(\varepsilon;\gamma_1,\dots, \gamma_n):=\{y\in H\colon d(x,P_{\gamma_i}y)<\varepsilon \ \forall\, i=1,2,\dots, n\}
\end{equation}
\tag{3.1}
$$
is contained in $U$. Here $P_{\gamma_i}y$ denotes the projection of $y$ onto the geodesic $\gamma_i$. Given a geodesic segment $\gamma\in\Gamma_x(H)$, we call $U_x(\varepsilon;\gamma)$ an elementary set. Proposition 3.1. The collection of weakly open sets $U$, together with the empty set $\varnothing$, defines a topology $\tau_w$ on $H$, and we call it the weak topology on $H$. Proof. It is clear that $H\in \tau_w$, and the empty set $\varnothing$ is in $\tau_w$ by definition. Let $\{U_i\}_{i\in I}$, where $I$ is some index, be weakly open for all $i\in I$. Take $x\in \bigcup_{i\in I}U_i$; then $x\in U_j$ for some $j\in I$. Since $U_j$ is weakly open, there exist $\varepsilon>0$ and $\gamma_1,\dots,\gamma_n\in\Gamma_x(H)$ such that $U_x(\varepsilon;\gamma_1,\dots,\gamma_n)\subseteq U_j\subseteq \bigcup_{i\in I}U_i$. Hence $\bigcup_{i\in I}U_i$ is weakly open. Now let $U_1$ and $U_2$ be two weakly open sets, and let $x\in U_1\cap U_2$. Then there exist $\varepsilon_1,\varepsilon_2>0$ and geodesic segments $\gamma_1,\dots,\gamma_n,\eta_1,\dots,\eta_m\in\Gamma_x(H)$ such that $U_{x}(\varepsilon_1;\gamma_1,\dots,\gamma_n)\subseteq U_1$ and $U_{x}(\varepsilon_2;\eta_1,\dots,\eta_m)\subseteq U_2$. Let $\varepsilon:=\min\{\varepsilon_1,\varepsilon_2\}$ and consider the set $U_x(\varepsilon; \gamma_1,\dots,\gamma_n,\eta_1,\dots,\eta_m)$. By construction it follows that Therefore, $U_1\cap U_2$ is weakly open. The proposition is proved. We show that any set $U$ is open in the usual metric topology. Lemma 3.1. Let $(H,d)$ be a Hadamard space. Then $U_{x}(\varepsilon;\gamma)$ is open in the metric topology for all $x\in H$ and $\gamma\in\Gamma_x(H)$. Proof. By inequality (2.3) the projections $P_C$ onto closed convex sets $C$ are nonexpansive mappings, that is, $d(P_{C}x,P_Cy)\leqslant d(x,y)$ for all $x,y\in H$. In particular, $P_{\gamma}$ is nonexpansive, that is, $d(P_{\gamma}x,P_{\gamma}y)\leqslant d(x,y)\hspace{0.2cm}\forall x,y\in H$. Now let $y\in U_{x}(\varepsilon;\gamma)$. Then $s:=d(x,P_{\gamma}y)$ satisfies $s<\varepsilon$. Therefore, the open geodesic ball $B(y,\varepsilon-s):=\{z\in H\colon d(x,y)<\varepsilon-s\}$ is contained in $U_{x}(\varepsilon;\gamma)$. Indeed, let $z\in B(y,\varepsilon-s)$. Then Therefore, $z \in U_{x}(\varepsilon;\gamma)$. The lemma is proved. We introduce the following notion, which plays a central role in this work. Definition 3.1. We call a Hadamard space weakly proper if every $x\in H$ is an interior point for every elementary set $U_x(\varepsilon;\gamma)$ in the weak topology $\tau_w$, that is, if for every $x\in H$ and every $\gamma \in \Gamma_{x}(H)$ there exists $V\in \tau_w$ such that $x \in V\subseteq U_x(\varepsilon;\gamma)$. Notice that weak properness does not imply that elementary sets are weakly open; however, $H$ is weakly proper whenever all elementary sets are weakly open. If $(H_1,d_1)$ and $(H_2,d_2)$ are weakly proper, then so is the product space $(H_1\times H_2,d)$ equipped with the product metric $d^2=d_1^2+d_2^2$. Moreover, that $x_n\to x$ yields $x_n\xrightarrow{w} x$ is obvious, for if $\lim_{n\to \infty}d(x,x_n)=0$, then the relation $d(x,P_{\gamma}x_n)=d(P_{\gamma}x,P_{\gamma}x_n)\leqslant d(x,x_n)$ implies that $\lim_{n\to \infty}d(x,P_{\gamma}x_n)=0$ for all $\gamma\in\Gamma_x(H)$. A sequence $(x_n)\subseteq H$ converges sequentially in $\tau_w$ to an element $x\in H$, and we denote this by $x_n\xrightarrow{\tau_w} x$, if and only if for every weakly open set $U$ containing $x$ all but finitely many elements of $(x_n)_{n\in\mathbb{N}}$ are in $U$. Theorem 3.1. Let $(x_n)_{n\in\mathbb{N}}\subset H$, and let $x\in H$. If $x_n\xrightarrow{w} x$, then $x_n\xrightarrow{\tau_w} x$. Moreover, if the Hadamard space is weakly proper, then $x_n\xrightarrow{\tau_w} x$ implies that $x_n\xrightarrow{w} x$. Proof. Let $x_n\xrightarrow{w} x$. Then for all $\gamma\in \Gamma_x(H)$ we have $\lim_{n\to \infty}d(x,P_{\gamma}x_n)=0$ or, equivalently, $x_n\in U_x(\varepsilon;\gamma)$ for all sufficiently large $n$. Let $U\in \tau_w$ contain $x$. Then there exist $\gamma_1,\dots,\gamma_n\in\Gamma_x(H)$ and $\varepsilon>0$ such that $U_x(\varepsilon;\gamma_1,\dots,\gamma_n)\subseteq U$. Since $x_n\in U_x(\varepsilon;\gamma_1,\dots,\gamma_n)$ for all sufficiently large $n$, we obtain $x_n\in U$ for all sufficiently large $n$.
Let $H$ be weakly proper. Suppose that $x_n\xrightarrow{\tau_w} x$ but $x_n\overset{w}\nrightarrow x$. Then there exists $\gamma\in\Gamma_x(H)$ and $\varepsilon>0$ such that $x_n\notin U_x(\varepsilon;\gamma)$ for infinitely many $n$. Weak properness implies that there is an open set $V\in\tau_w$ containing $x$ such that $V\subseteq U_x(\varepsilon;\gamma)$. Therefore, $x_n\notin V$ for infinitely many $n$. However, this contradicts the convergence $x_n\xrightarrow{\tau_w} x$. The theorem is proved. It is clear from Theorem 3.1 that $x_n\to x$ implies that $x_n\xrightarrow{\tau_w} x$. Proof. Let $x,y\in H$ be two distinct points and $\gamma, \widetilde \gamma\colon [0,1]\to H$ be the geodesics connecting $x$ with $y$ such that $\gamma(0)=x$, $\gamma(1)=y$ and $\widetilde \gamma(0)=y$, $\widetilde \gamma(1)=x$. Let $l_{\gamma}:=d(x,y)$, and let $\varepsilon\in (0,l_{\gamma})$. Note that $l_{\gamma}=l_{\widetilde\gamma}$ and $P_{\gamma}z=P_{\widetilde\gamma}z$ for all ${z\in H}$. Define the sets $U_x(\varepsilon; \gamma):=\{z\in H\colon d(x,P_{\gamma}z)<\varepsilon\}$ and $U_y(l_{\gamma}-\varepsilon; \widetilde\gamma):=\{{z\in H}\colon d(y,P_{\widetilde\gamma}z)<l_{\gamma}-\varepsilon\}$. Suppose there is some $z_0\in U_x(\varepsilon; \gamma)\cap U_y(l_{\gamma}-\varepsilon; \widetilde\gamma)$; then $d(x,P_{\gamma}z_0)<\varepsilon$ and $d(y,P_{\widetilde\gamma}z_0)<l_{\gamma}-\varepsilon$ would imply that $l_{\gamma}=d(x,y)\leqslant d(x,P_{\gamma}z_0)+d(y,P_{\gamma}z_0)= d(x,P_{\gamma}z_0)+d(y,P_{\widetilde\gamma}z_0)<l_{\gamma}$ which is impossible. Therefore, $U_x(\varepsilon; \gamma)\cap U_y(l_{\gamma}-\varepsilon; \widetilde\gamma)=\varnothing$. Since $(H,d)$ is weakly proper, there are $V,W\in\tau_w$ such that $x\in V\subseteq U_x(\varepsilon; \gamma)$ and $y\in W\subseteq U_y(l_{\gamma}-\varepsilon; \widetilde\gamma)$. Then $V\cap W\subseteq U_x(\varepsilon; \gamma)\cap U_y(l_{\gamma}-\varepsilon; \widetilde\gamma)=\varnothing$. The proof is complete. 3.2. Convex sets and compactnessWe say that a set $S\subseteq H$ is weakly closed if it is closed with respect to $\tau_w$. Theorem 3.2. Let $(H,d)$ be a Hadamard space. A convex set $C\subseteq H$ is strongly closed if and only if it is weakly closed. Proof. Any weakly closed set is strongly closed. So let $C$ be a strongly closed convex set. We show that $C$ is weakly closed. It suffices to show that $H\setminus C$ is weakly open. Let $y\in H\setminus C$. Then $P_Cy$ exists and is unique. Let $\gamma\colon [0,1]\to H$ be the geodesic connecting $y$ with $P_Cy$ such that $\gamma(0)=y$ and $\gamma(1)=P_Cy$. For $\varepsilon\in(0,l(\gamma))$, where $l(\gamma):=d(y,P_Cy)$ is the length of the geodesic $\gamma$, consider the elementary set $U_y(\varepsilon;\gamma)$. We need only to show that $U_y(\varepsilon;\gamma)\cap C=\varnothing$. Let $x\in C$, and let $z\in H$ be arbitrary. Since both $C$ and $\gamma$ are strongly closed and convex, an application of inequality (2.3) yields $d(x,z)^2\geqslant d(x,P_{C}z)^2+d(P_Cz,z)^2$ and $d(x,P_Cy)^2\geqslant d(x,P_{\gamma}x)^2+d(P_{\gamma}x,P_Cy)^2$, where we have used that $x$ lies in the convex set $C$ for the first inequality and that $P_Cy$ lies in the convex set $ \gamma$ for the second inequality. Now let $z=P_\gamma x$ be the projection of $x$ onto $\gamma$. Since $z \in \gamma$, we have $P_C z= P_C y$; see [5]. Then the above two inequalities imply that $P_Cy = z= P_{\gamma}x$. As $d(y,P_{\gamma}x)=d(y,P_Cy)>\varepsilon$ for all $x\in C$, it follows that $U_y(\varepsilon;\gamma)\cap C=\varnothing$, that is, $U_y(\varepsilon;\gamma)\in H\setminus C$. The proof is complete. Theorem 3.3 (Mazur’s lemma). Let $(x_n)\subseteq H$ be a sequence such that $x_n\xrightarrow{w} x$ for some $x\in H$. Then there exists a function $N\colon \mathbb{N}\to\mathbb{N}$ and a sequence $(y_n)\subseteq H$ such that $y_n\to x$ and $y_n\in \operatorname{co}(\{x_1,x_{2},\dots,x_{N(n)}\})$ for all $n\in\mathbb{N}$. Proof. Weak convergence $x_n\xrightarrow{w} x$ implies that $x\in \operatorname{wcl}\{x_1,x_2,\dots\}$ by Theorem 3.1, where $\operatorname{wcl}\{x_1,x_2,\dots\}$ is the weak closure of $\{x_1,x_2,\dots\}$. Moreover, $\{x_1,x_2,\dots\} \subseteq\operatorname{co}(\{x_1, x_2,\dots\})$ yields $\operatorname{wcl}\{x_1,x_2,\dots\} \subseteq\operatorname{wcl}\operatorname{co}(\{x_1,x_2,\dots\})$, hence we have $x\in \operatorname{wcl}\operatorname{co}(\{x_1,x_2,\dots\})$. The strong closure $\operatorname{cl}\operatorname{co}(\{x_1,x_2,\dots\})$ of the convex set $\operatorname{co}(\{x_1,x_2,\dots\})$ is a closed convex set. Indeed, if $C$ is (geodesically) convex, then so is $\operatorname{cl} C$. In order to see this, let $u,v \in \operatorname{cl} C$. Consider sequences $(u_n)$ and $(v_n)$ in $C$ such that $u_n \to u$ and $v_n\to v$. Let $\gamma$ be the geodesic connecting $u$ with $v$, and let $\gamma_n$ be the geodesics connecting $u_n$ with $v_n$. From the characteristic inequality (2.4) we obtain $d(\gamma(t),\gamma_n(t)) \to 0$ as $n\to \infty$. Hence $\operatorname{cl} C$, and therefore $\operatorname{cl}\operatorname{co}(\{x_1,x_2,\dots\})$, are indeed convex. By Theorem 3.2, $\operatorname{cl}\operatorname{co}(\{x_1,x_2,\dots\})$ is weakly closed. It follows that $x\in \operatorname{wcl}\operatorname{co}(\{x_1,x_2,\dots\}) \subseteq\operatorname{cl}\operatorname{co} (\{x_1,x_2,\dots\})$. Then there exists some sequence $(y_n)\!\subseteq\! \operatorname{co} (\{x_1,x_2,\dots\})$ such that $y_n\!\to\! x$. Additionally, we have $\operatorname{co}(\{x_1,x_2,\dots\}) =\bigcup_{k\in\mathbb{N}}\operatorname{co}(\{x_1,x_{2},\dots,x_{k}\})$. Hence $y_n\in \operatorname{co}(\{x_1,x_{2},\dots,x_{k(n)}\})$ for some $k(n)$. For each $n$ set $N(n):=k(n)$. The proof is complete. Next we turn to compactness properties with respect to weak convergence and the weak topology $\tau_w$. A set $K\subseteq H$ is called weakly sequentially compact if every sequence in $K$ has a weakly convergent subsequence. Similarly, $K$ is called $\tau_w$-sequentially compact if every sequence in $K$ has a $\tau_w$-convergent subsequence. Weak sequential compactness implies $\tau_w$-sequential compactness, and these notions coincide on a weakly proper Hadamard space. Finally, a set $K$ is called weakly compact if for any open cover in $\tau_w$ of $K$ there is a finite subcover. Lemma 3.3 (see [7], Proposition 3.1.2). Every bounded sequence in a Hadamard space has a weakly convergent subsequence. Lemma 3.4 (see [7], Lemma 3.2.1). Let $K\subseteq H$ be a closed convex set, and let $(x_n)_{n\in\mathbb{N}}\subset K$. If $x_n\xrightarrow{w} x$, then $x\in K$. Lemmas 3.3 and 3.4 immediately imply the following result. Theorem 3.4. Any bounded closed convex set $K$ in a Hadamard space is weakly sequentially compact and therefore $\tau_w$-sequentially compact. Theorem 3.5. Let $(H,d)$ be separable. Then any weakly sequentially compact set $K \subset H$ is weakly compact. Proof. The proof proceeds by contradiction. Suppose that $K\subseteq H$ is weakly sequentially compact but not weakly compact. Then there exists some open cover $\{U_i\}_{i\in I}$ of $K$ in $\tau_w$ that has no finite subcover. By assumption $(H,d)$ is separable. Hence $(H,d)$ is a Lindelöf space, that is, every open cover (in the strong topology) has a countable subcover; see Theorem 6.7 in [18], for example. Since $\{U_i\}_{i\in I}$ is an open cover in the usual metric topology, there exists a countable subcover $\{U_j\}_{j\in J}$. Let $V_n:=\bigcup_{j=1}^nU_{j}$. Then $W_n:=H\setminus V_n$ is weakly closed for all $n$. Moreover, the family of sets $W_n$ satisfies $W_{n+1}\subseteq W_n$. Because $V_n$ cannot cover $K$, we have that $W_n\cap K$ is nonempty for every $n\in\mathbb{N}$. Let $x_n\in W_n\cap K$. Since $K$ is weakly sequentially compact and, consequently, $\tau_w$-sequentially compact, the sequence $(x_n)$ has a subsequence $(x_{n_k})$ that converges in $\tau_w$ to some element $x^*\in K$. Let $\mathcal{U}_w(x^*)$ denote the collection of weakly open sets containing $x^*$. Then for each $U\in \mathcal{U}_w(x^*)$ and for each $n\in \mathbb{N}$ there exists $m\geqslant n$ such that $U\cap W_m\neq\varnothing$ and, in particular, $U\cap W_n\neq\varnothing$, implying that $x^*\in\operatorname{wcl} W_n=W_n$. Since $n$ is arbitrary, we have $x^*\in\bigcap_nW_n$. Hence $x^*\in \bigcap_nW_n\cap K$. Therefore, $\bigcap_{n\in\mathbb{N}}W_n\cap K\neq\varnothing$, which, together with $\bigcap_{n\in\mathbb{N}}W_n\cap K\subsetneq K$, yields that $K\supsetneq K\setminus( \bigcap_{n\in\mathbb{N}}W_n\cap K)=K\setminus\bigcap_{n\in\mathbb{N}}(K\setminus V_n)=\bigcup_{n\in\mathbb{N}}(K\cap V_n)=K$. The theorem is proved. Remark 3.1. Notice that the previous proof also applies to the more general setting of a topology that is weaker than the metric topology in any separable metric space. Proposition 3.2. Let $(H,d)$ be a Hadamard space that is Hausdorff with respect to $\tau_w$, and let $K\subseteq H$ be bounded. If $K$ weakly compact, then $K$ is weakly sequentially compact. Proof. Let $(x_n)_{n\in\mathbb N}\subseteq K$, then $(x_n)$ is bounded. By Lemma 3.3 there is a subsequence $(x_{n_k})\xrightarrow{w} x$ for some $x\in H$. By Theorem 3.1 we have $x_{n_k}\xrightarrow{\tau_w} x$. The assumption that $\tau_w$ is Hausdorff implies that $K$ is weakly closed and, consequently, $\tau_w$-sequentially closed, hence $x\in K$. The proposition is proved. As an immediate consequence of Theorem 3.5 and Proposition 3.2, we obtain the following. Corollary 3.1. Let $(H,d)$ be separable and Hausdorff with respect to $\tau_w$. If $K\subseteq H$ is bounded, then $K$ is weakly compact if and only if $K$ is weakly sequentially compact. Example 3.1. This question is motivated by the example of a simplicial tree given by Monod [20]. This tree consists of countably many rays of finite but ever increasing length, all meeting at one vertex. If $(V_{i})_{i\in I}$ is some open cover in $\tau_w$ for the simplicial tree then there exists some $i\in I$ such that $V_i$ contains an elementary set that contains the common vertex. By construction this elementary set covers most of the space except for a certain ray which can be covered by at most a finite number of elementary sets. Therefore, we can always find a finite subcover of our original cover $(V_{i})_{i\in I}$, that is, the simplicial tree is weakly compact. Moreover, this space also provides an example of an unbounded weakly convergent sequence; see the discussion in § 3.1.1 of [3]. However, we can give a conditional answer to Question 3.1. First we need a regularity assumption. Let $(H,d)$ be separable and $\{y_n\}_{n\in\mathbb N}$ be a dense set in $H$. We say that $(H,d)$ has the nice geodesics structure if for any $x\in H$ and any sequence $(x_k)_{k\in\mathbb N}$ such that $\lim_kP_{[x,y_n]}x_k=x$ for all $n\in\mathbb N$ we have $\lim_kP_{\gamma}x_k=x$ for any $\gamma\in \Gamma_x(H)$. Theorem 3.6. Let $(H,d)$ be separable and weakly proper space enjoying the nice geodesics structure. Then a weakly compact set is weakly sequentially compact. Proof. Let $(H,d)$ be weakly proper and $K\subseteq H$ be a weakly compact set. Let $(x_k)_{k\in\mathbb{N}}$ be a sequence in $K$. Then by virtue of Theorem 3.15 in [3] $(x_k)_{k\in\mathbb{N}}$ has a weak accumulation point $x\in K$, hence by Proposition 3.14 in [3] $x$ is also a weak limit point. Since $H$ is separable, there exits a dense countable set $\{y_n\}$ in $H$. Let $\gamma_n\colon [0,1]\to H$ denote the geodesic connecting $x$ with $y_n$ for every $n\in\mathbb{N}$. Now consider the family of sets $V_n$ defined by Since $(H,d)$ is weakly proper, there are $U_{n,i}\in\tau_w$ containing $x$ such that $U_{n,i}\subseteq U_x(1/n;\gamma_i)$ for every $i=1,2,\dots,n$ and for every $n\in\mathbb{N}$. Set $U_n:=\bigcap_{i=1}^nU_{n,i}$; then $U_n\subseteq V_n$ for all $n\in\mathbb{N}$. Since $U_n$ is a weakly open set containing $x$, it has at least one element $x_{k(n)}$. Passing to a subsequence if necessary, we can say that $x_n \in U_n$ for all $n$. In particular $x_n\in V_n$ for all $n$. The sets $V_n$ are nested, so we have $x_m \in V_n$ for all $m \geqslant n$, that is, $\lim_mP_{\gamma_i}x_m=x$ for all $i=1,2,\dots,n$ and thus $\lim_mP_{\gamma_n}x_m=x$ for all $n\in\mathbb{N}$. Because $(H,d)$ enjoys the nice geodesics structure, we have $\lim_mP_{\gamma}x_m=x$ for all $\gamma\in\Gamma_x(H)$. The theorem is proved. Corollary 3.3. In a separable and weakly proper Hadamard space that enjoys the nice geodesics structure weak compactness and weak sequential compactness coincide. In relation to the above observations we ask the following. Another condition equivalent to the nice geodesics structure is as follows. Proposition 3.3. Let $(H,d)$ be separable, and let $\{y_n\}_{n\in\mathbb N}\subset H$ be dense in $H$. Then $(H,d)$ has the nice geodesics structure if and only if for every elementary set $U_x(\varepsilon;\gamma)$ there exists a finite family $\{y_{n(i)}\}_{i=1}^N\subset\{y_n\}_{n\in\mathbb N}$ and $\delta>0$ such that $\bigcap_{i=1}^NU_x(\delta;\gamma_i)\subseteq U_x(\varepsilon;\gamma)$, where $\gamma_i=[x,y_{n(i)}]$ for $i=1,2,\dots,N$. Proof. Let $(H,d)$ have the nice geodesics structure. Suppose that there is an elementary set $U_x(\varepsilon;\gamma)$ such that for no finite family $\{y_{n(i)}\}_{i=1}^N\subset\{y_n\}_{n\in\mathbb N}$ and no $\delta>0$ the inclusion $\bigcap_{i=1}^NU_x(\delta;\gamma_i)\subseteq U_x(\varepsilon;\gamma)$ holds. Letting $V_n$ be defined as in (3.2) we can construct a sequence $(x_n)_{n\in\mathbb N}\subset H\setminus U_x(\varepsilon;\gamma)$ such that $\lim_{n\to+\infty}P_{\gamma_k}x_n=x$ along every segment $\gamma_k=[x,y_k]$ for $k\in\mathbb N$. Then the nice geodesics structure implies that $\lim_{n\to+\infty}P_{\gamma}x_n=x$, that is, $x_n\in U_x(\varepsilon;\gamma)$ for all sufficiently large $n\in\mathbb N$. This is impossible.
Now suppose that for every elementary set $U_x(\varepsilon;\gamma)$ there exists a finite family $\{y_{n(i)}\}_{i=1}^N\subset\{y_n\}_{n\in\mathbb N}$ and $\delta>0$ such that $\bigcap_{i=1}^NU_x(\delta;\gamma_i)\subseteq U_x(\varepsilon;\gamma)$, where $\gamma_i=[x,y_{n(i)}]$ for $i=1,2,\dots,N$. Let $(x_n)_{n\in\mathbb N}\subset H$ be a sequence such that $P_{\gamma_k}x_n=x$ along all $\gamma_k=[x,y_k]$. Then $x_n\in \bigcap_{i=1}^NU_x(\delta;\gamma_i)$ for all sufficiently large $n\in\mathbb N$, that is, $x_n\in U_x(\varepsilon;\gamma)$ for all sufficiently large $n\in\mathbb N$. Since $\varepsilon>0$ is arbitrary, we have $\lim_{n\to+\infty}P_{\gamma}x_n=x$. This completes the proof. 3.3. Locally compact spacesWe now turn to locally compact Hadamard spaces. Recall that a topological space $(X,\tau)$ is said to be locally compact if for every $x\in X$ there exists an open set $U\in \tau$ and a compact set $K$ such that $x\in U\subseteq K$. Theorem 3.7 (see [13], Proposition 4.3). The followings statements are equivalent for a Hadamard space $(H,d)$: Lemma 3.5. Let $(H,d)$ be locally compact Hadamard space and $(x_{\alpha})_{\alpha\in\mathcal A}$ be a net in $H$. If $x_{\alpha}\xrightarrow{w} x$, then $(x_{\alpha})$ is bounded. Proof. Let $x_{\alpha}\xrightarrow{w} x$, but suppose that $(x_{\alpha})$ is unbounded. Then without loss of generality we can assume that $d(x_{\alpha},x) \geqslant R>0$ for all $\alpha\in \mathcal{A}$. Consider the closed geodesic ball $C:=\mathbb{B}(x,R)$. Denote by $P_{C}x_{\alpha}$ the projection of $x_{\alpha}$ onto $C$ for every $\alpha\in \mathcal{A}$. By assumption $(H,d)$ is locally compact. Theorem 3.7 implies that $C$ is compact and, in particular, the boundary $\partial C$ is compact. Hence $(P_Cx_{\alpha})$ is a net in the compact set $\partial C$. By Theorem 2 in [15], § 5, there exits a subnet $(P_Cx_{\beta})_{\beta\in\mathcal B}$ converging to some element $z\in\partial C$. Let $\gamma\colon [0,1]\to H$ denote the geodesic segment connecting $x$ with $z$. Clearly, $\gamma\subset C$. For each $\beta\in\mathcal B$ denote by $\gamma_{\beta}\colon [0,1]\to H$ the geodesic segment connecting $x$ with $P_Cx_{\beta}$. Let $P_{\gamma}x_{\beta}$ denote the projection of $x_{\beta}$ onto the geodesic segment $\gamma$. From the triangle inequality we obtain $d(P_Cx_{\beta},z)\geqslant |d(x_{\beta},P_Cx_{\beta})-d(x_{\beta},z)|$, which implies that $\lim_{\beta}|d(x_{\beta},P_Cx_{\beta})-d(x_{\beta},z)|=0$. Since both $C$ and $\gamma$ are strongly closed and convex, we have the quadratic inequalities $d(x_{\beta},P_Cx_{\beta})^2+d(P_Cx_{\beta},P_{\gamma}x_{\beta})^2\leqslant d(x_{\beta},P_{\gamma}x_{\beta})^2$ and $d(x_{\beta},P_{\gamma}x_{\beta})^2+d(P_{\gamma}x_{\beta},z)^2\leqslant d(x_{\beta},z)^2$, implying that $d(x_{\beta},P_Cx_{\beta})\leqslant d(x_{\beta},P_{\gamma}x_{\beta})\leqslant d(x_{\beta},z)$. Therefore, we have
$$
\begin{equation}
\lim_{\beta}|d(x_{\beta},P_{\gamma}x_{\beta})-d(x_{\beta},P_Cx_{\beta})|=0.
\end{equation}
\tag{3.3}
$$
By assumption $x_{\alpha}\xrightarrow{w} x$ and, in particular, $x_{\beta}\xrightarrow{w} x$. Therefore, it follows that $P_{\gamma}x_{\beta}\to x$. Consider the geodesic segment $\eta_{\beta}\colon [0,1]\to H$ connecting $x$ with $x_{\beta}$. Then there exists $z_{\beta}\in\eta_{\beta}$ such that $z_{\beta}\in\partial C$ for every $\beta\in\mathcal B$. Since $z_{\beta}\in\partial C$, we obtain $d(x_{\beta},P_Cx_{\beta})\leqslant d(x_{\beta},z_{\beta})$ and thus $d(x_{\beta},x)=d(x_{\beta},z_{\beta})+d(z_{\beta},x)\geqslant d(x_{\beta},P_Cx_{\beta})+R$ for each $ \beta\in\mathcal B$ which in turn implies that $|d(x_{\beta},x)-d(x_{\beta},P_Cx_{\beta})|\geqslant R\!>\!0$. By the triangle inequality we have again $d(P_{\gamma}x_{\beta},x)\!\geqslant\!|d(x_{\beta},x)-d(x_{\beta}, P_{\gamma}x_{\beta})|$, for each $ \beta\in\mathcal B$. Therefore, the equality $\lim_{\beta}P_{\gamma}x_{\beta}=x$ implies that $\lim_{\beta}|d(x_{\beta},x)-d(x_{\beta}, P_{\gamma}x_{\beta})|=0$. Moreover, we have
$$
\begin{equation*}
\begin{aligned} \, 0<R &\leqslant |d(x_{\beta},x)-d(x_{\beta},P_Cx_{\beta})| \\ &\leqslant |d(x_{\beta},x)-d(x_{\beta}, P_{\gamma}x_{\beta})|+|d(x_{\beta},P_{\gamma}x_{\beta}) -d(x_{\beta},P_Cx_{\beta})|, \end{aligned}
\end{equation*}
\notag
$$
which, together with (3.3), yields a contradiction since the right-hand side vanishes in the limit. The lemma is proved. Corollary 3.4. Let $(H,d)$ be a locally compact Hadamard space and $(x_{n})_{n\in\mathbb N}$ be a sequence in $H$. If $x_{n}\xrightarrow{w} x$, then $(x_{n})$ is bounded. Proof. Any weakly open set is open. Now let $U$ be an open set, and suppose that $U$ is not weakly open. Then there is $x\in U$ such that for any finite collection of geodesic segments $\gamma_1,\gamma_2,\dots,\gamma_n\in\Gamma_x(H)$ and any $\varepsilon>0$ the set $\bigcap_{i=1}^nU_x(\varepsilon;\gamma_i)$ is not entirely contained in $U$. Let $\mathscr F$ be the collection of all finite subsets of $\Gamma_x(H)$. Two sets $F_1,F_2\in\mathscr F$ are said to be equivalent whenever $|F_1|=|F_2|$. Let $[F]$ denote an equivalence class and $|[F]|=n_F$ its cardinality. Consider the sets $V_F=\{{y\in C}\colon d(x,P_{\gamma}y)<1/n_F,\,\gamma\in F\}$. By the above observation, for every $F\in\mathscr F$ there exists $x_F\in V_{F}\setminus U$. If we set $F_1\geqslant F_2$ whenever $F_1\subseteq F_2$, then $(\mathscr F,\geqslant)$ is a directed set. In particular, $(x_F)_{F\in\mathscr F}$ is a net, and by construction $(x_F)_{F\in\mathscr F}$ converges weakly to $x$. By Lemma 3.5 the net $(x_F)_{F\in\mathscr F}$ is bounded and, consequently, by Theorem 3.7 the (strong) closure $\operatorname{cl}(\{x_{F}\}_{F\in\mathscr F})$ is compact in $H$. Then by virtue of Theorem 2 in [15], § 5, there is a subnet $(x_{F'})_{F'\in\mathscr F'}$ of the net $(x_F)$ converging to an element $y\in \operatorname{cl}(\{x_{F}\}_{F\in\mathscr F})$. In particular, $x_{F'}\xrightarrow{w} y$. By Remark 2.1, as applied to nets, we have $y=x$. This is impossible since by construction $x_{F'}\notin U$ for all $F'\in\mathscr F'$. The theorem is proved. Proof. By Theorem 3.8 the elementary sets $U_x(\varepsilon;\gamma)$ are weakly open for every $x\in H, \gamma\in\Gamma_x(H)$ and $\varepsilon>0$. In particular, $x$ is an interior point in $\tau_w$ of any elementary set $U_x(\varepsilon;\gamma)$. 3.4. Another weak topologyWe discuss briefly Remark 3.28 in [3] about another topology which is somewhat different from $\tau_w$. By Lemma 3.1 the elementary sets $U_x(\varepsilon;\gamma)$ are open in the usual metric topology. Then the collection of sets $\{U_x(\varepsilon;\gamma)\colon x\in H,\, \gamma\in\Gamma_x(H)\}$, together with $H$, forms a subbasis for a topology $\tau$, which is weaker then the metric topology. Moreover, convergence in $\tau$ implies weak convergence (convergence along geodesics), for if $x_n\xrightarrow{\tau} x$, then for any $\gamma\in\Gamma_x(H)$ and any $\varepsilon>0$ we have $x_n\in U_x(\varepsilon;\gamma)$ for all sufficiently large $n$, that is, $\lim_{n\to \infty}d(x,P_{\gamma}x_n)=0$ for any $\gamma\in\Gamma_x(H)$; therefore, $x_n\xrightarrow{w} x$. In particular, $\tau_w\subseteq\tau$ and these topologies coincide if and only if the elementary sets $U_x(\varepsilon;\gamma)$ are open in $\tau_w$. The topology $\tau$ enjoys some properties without any additional assumptions on the space. For example, it is Hausdorff and a closed convex set is $\tau$-closed (the proofs of these claims can be identical to the ones for $\tau_w$). Moreover, in a locally compact space $\tau$ coincides with the metric topology. This topology and its relationships to other weak topologies are studied in more details in the subsequent work [19]. 3.5. The case of a Hilbert spaceLet $H$ be a Hilbert space $(\mathcal{H}, \|\cdot\|)$ equipped with its canonical norm. Note that for every $x\in \mathcal{H}$ each geodesic segment ${\gamma\!\in\!\Gamma_x(\mathcal{H})}$ corresponds to a unique straight line $l$ passing through $x$. Given $\gamma,\eta\in\Gamma_x(\mathcal{H})$, we say that $\gamma$ is equivalent to $\eta$, and write $\gamma\sim\eta$, if and only if $\gamma$ and $\eta$ belong to the same line $l$. Let $[l]$ denote the equivalence class of all geodesic segments $\gamma\in\Gamma_x(\mathcal{H})$ sharing the same line $l$. Our aim is to show that our notion of weak convergence along geodesics coincides with the usual weak convergence in a Hilbert space. Moreover, we prove that a Hilbert space is weakly proper. Lemma 3.6. A sequence $(x_n)\subset\mathcal{H}$ converges weakly (along geodesics) to $x\in\mathcal{H}$ if and only if $P_{l}x_n\to x$ as $n\uparrow+\infty$ for all lines $l$ containing $x$. Proof. Suppose that $\lim_nP_lx_n=x$ for all lines $l$ containing $x$. Let $\gamma\in\Gamma_x(\mathcal{H})$ be such that $\gamma\subset l$. Then all but finitely many points $P_lx_n$ lie in the image of $\gamma$, that is, $P_lx_n\,{=}\,P_{\gamma}x_n$ for all sufficiently large $n$. This means that ${\lim_nP_{\gamma}x_n\,{=}\,\lim_nP_lx_n\,{=}\,x}$. Since this holds for any $l$ containing $x$ and for any $\gamma\in[l]$, we have $\lim_nP_{\gamma}x_n=x$ for any geodesic segment $\gamma\in\Gamma_x(\mathcal{H})$. Now let $x_n\xrightarrow{w} x$. By definition $\lim_nP_{\gamma}x_n=x$ for all $\gamma\in\Gamma_x(\mathcal{H})$. Each $\gamma\in\Gamma_x(\mathcal{H})$ determines a unique line $l$ containing $x$. Then a similar argument shows that $\lim_nP_lx_n=x$ for all lines containing $x$. The lemma is proved. With the collection $\{\phi_{\gamma}(x; * )\}$, where $\phi_{\gamma}(x; y):=||x-P_{\gamma}y||$ for $ \gamma \in \Gamma_x(H)$ and for all $y\in \mathcal{H}$, we can associate the function $\phi_{[l]}(x;\,\cdot\,)$ defined by $\phi_{[l]}(x;y):=\|x-P_{l}y\|$. Let $\mathcal{H}^*$ denote the dual of the Hilbert space $\mathcal{H}$, that is the space of all bounded linear continuous functionals on $\mathcal H$. A sequence $(x_n)_{n\in\mathbb{N}}\subseteq \mathcal{H}$ is said to converge weakly (in the usual sense) to $x\in\mathcal{H}$ if and only if $\lim_nf(x_n)=f(x)$ for all $f\in \mathcal{H}^*$. Proposition 3.4. In a Hilbert space weak convergence (along geodesics) coincides with the usual notion of weak convergence. Proof. For simplicity we assume that $\mathcal{H}$ is a real Hilbert space. The case of a complex Hilbert space is dealt with accordingly by taking care separately of the real and imaginary parts of the inner product. Let $x\in \mathcal{H}$, and set $\mathcal{H}^*_x:=\{\phi_{\gamma}\colon\gamma\in\Gamma_x(\mathcal{H})\}$. From Lemma 3.6 it follows that $\lim_n\phi_{\gamma}(x;x_n)=0$ for all $\gamma\in[l]$ if and only if $\lim_n\phi_{[l]}(x;x_n)=0$. Therefore, it is sufficient to limit ourselves to the family of functions $\{\phi_{[l]}(x;\,\cdot\,)\}_L$, where $L$ is the set of all lines containing $x$. Clearly, $\{\phi_{[l]}(x;\,\cdot\,)\}_L=\mathcal{H}^*_x/{\sim}$. Given $y\in\mathcal{H}$ and $z_l\in l$, let $\theta$ be the angle between the vectors $y-x$ and $z_l-x$. Then from the cosine formula for inner products we obtain $\langle y-x, z_l-x\rangle=\|y-x\|\|z_l-x\|\cos\theta$. Realizing that $\|y-x\|\cos\theta=\pm\|x-P_ly\|$, it follows that Using the linearity of the inner product one can write the last equality as Given $x\in\mathcal{H}$, two elements $z_1,z_2\in\mathcal{H}$ are equivalent, $z_1\sim_x z_2$, if and only if $z_1$ and $z_2$ lie on the same line $l$ passing through $x$. It is known that $\mathcal{H}^*$ is isometrically isomorphic to $\mathcal{H}$. There is a one-to-one correspondence between $\mathcal{H}_x^*/{\sim}$ and $\mathcal{H}^*/{\sim_x}$ via the mapping $\mathcal{H}^*/{\sim_x}\ni [z]\mapsto \{\pm u_l\}\mapsto\phi_{[l]}\in\mathcal{H}_x^*/{\sim}$. Moreover, it follows from the above equality that weak convergence (along geodesics) coincides with the usual notion of weak convergence. The proposition is proved. Proposition 3.5. In a Hilbert space elementary sets are weakly open. Thus, a Hilbert space is weakly proper, and the usual weak topology coincides with $\tau_w$. Proof. Let $x\in\mathcal{H}$ and consider the elementary set $U_x(\varepsilon;\gamma)$ for some $\varepsilon>0$ and $\gamma\in\Gamma_x(\mathcal{H})$. Let $l$ be the unique straight line passing through $x$ and corresponding to the geodesic segment $\gamma$. Let $y\in U_x(\varepsilon;\gamma)$, and set $l':=l+(y-x)$. Then $l'$ is parallel to $l$. Let $\alpha$ denote the plane determined by $l$ and $l'$. Note that for any $z\in \mathcal{H}$ the projections $P_{\alpha}z$, $P_{l}z$ and $P_{l'}z$ are collinear and, moreover, their common line $l''$ is perpendicular to both $l$ and $l'$. This argument, combined with $y\in U_x(\varepsilon;\gamma)$, implies that for some $\delta>0$ and $\gamma'\in\Gamma_y(\mathcal{H})$ belonging to $[l']$ we have $U_y(\delta;\gamma')\subseteq U_x(\varepsilon;\gamma)$. Because $\phi^{-1}_{[\ell]}([0,\varepsilon))=U_x(\varepsilon;[\ell])$ and the correspondence between $\mathcal{H}_x^*/{\sim}$ and $\mathcal{H}^*/{\sim_x}$ is one-to-one, the usual weak topology in $\mathcal H$ coincides with $\tau_w$. The proposition is proved. From Corollary 3.5 and Proposition 3.5 we have the following. Corollary 3.7. The Cartesian product of a locally compact Hadamard space and a Hilbert space is weakly proper. Remark 3.2. The simplicial tree is an example of a non-Hilbertian weakly proper space that is not the Cartesian product of a locally compact and a Hilbert space. § 4. Other forms of weak topologyThere were attempts previously to identify the weak topologies corresponding to some notions of weak convergence in Hadamard spaces. These attempts have offered other perspectives on this topic, which we compare to our notion. 4.1. The Kakavandi weak topologyKakavandi [13] proposed a notion of a weak topology which is based on the following observation. In a Hilbert space $(\mathcal{H},\|\,{\cdot}\,\|)$ equipped with its canonical norm $\|\,{\cdot}\,\|$ a sequence $(x_n)$ converges weakly to an element $x\in \mathcal{H}$ if and only if $\lim_{n\to \infty}\langle x_n,y\rangle=\langle x,y\rangle$ for all $y\in \mathcal{H}$. This is equivalent to $ \lim_{n\to \infty}\langle x_n-z,y-z\rangle=\langle x-z,y-z\rangle$ for all $y,z\in \mathcal{H}$. Then the identity
$$
\begin{equation}
\langle x-z,y-w\rangle=\frac{1}{2}(|x-y|^2+|z-w|^2-|x-w|^2-|z-y|^2)
\end{equation}
\tag{4.1}
$$
gives rise to the possibility of extending the definition of weak convergence to a general metric space $(X,d)$ by expressing the right-hand side of (4.1) in terms of the metric $d(\,\cdot\,{,}\,\cdot\,)$. Following Berg and Nikolaev [4], consider the Cartesian product $X\times X$ where $X$ is a general metric space. Each pair $(x,y)\in X\times X$ determines a so-called bound vector which is denoted by $\overrightarrow{xy}$. The point $x$ is called the tail of $\overrightarrow{xy}$ and $y$ is called its head. The zero bound vector is $\overrightarrow{0}_x=\overrightarrow{xx}$. The length of a bound vector $\overrightarrow{xy}$ is defined as the metric distance $d(x,y)$. Furthermore, if $\overrightarrow{u}:=\overrightarrow{xy}$, then $-\overrightarrow{u}:=\overrightarrow{yx}$. Let
$$
\begin{equation}
\langle\overrightarrow{xz},\overrightarrow{yw}\rangle :=\frac{1}{2}(d(x,y)^2+d(z,w)^2-d(x,w)^2-d(z,y)^2).
\end{equation}
\tag{4.2}
$$
Kakavandi’s notion of weak convergence is defined in the following way. A sequence $(x_n)$ in a Hadamard space $(H,d)$ converges weakly to an element $x\in H$ if and only if $\lim_{n\to \infty}\langle\overrightarrow{xx_n},\overrightarrow{xy}\rangle=0$ for all $y\in H$. This form of weak convergence coincides with the usual weak convergence in a Hilbert space. It turns out, however, that Kakavandi’s notion of convergence does not coincide with $\Delta$-convergence; see Example 4.7 in [13]. There is a natural topology associated with Kakavandi convergence and generated by the sets of the form
$$
\begin{equation}
W(x,y;\varepsilon):=\{z\in H\colon |\langle\overrightarrow{xz},\overrightarrow{xy}\rangle|<\varepsilon\}, \qquad x,y\in H, \quad \varepsilon>0.
\end{equation}
\tag{4.3}
$$
More precisely, the family of sets $\{W(x,y;\varepsilon)\colon x,y\in H,\,\varepsilon>0\}$ forms a subbasis for the Kakavandi topology $\tau_K$. One has $x_n\xrightarrow{K} x$ if and only if $x_n\xrightarrow{\tau_K} x$; see [13], Theorem 3.2. Moreover, the Kakavandi topology is Hausdorff. Proof. To show (i) let $U\in\tau_w$ be a weakly open set, and let $x\in U$. By the construction of the topology $\tau_w$ there exist a finite number of geodesic segments $\{\gamma_i\}_{i=1}^n$ starting at $x$ such that $\bigcap_{i=1}^nU_x(\delta;\gamma_i)\subset U$. For simplicity suppose $n=1$, that is, $U_x(\delta;\gamma)\subset U$ for some $\gamma$ starting at $x$. Let $y\in\gamma$ be such that $d(x,y)=\delta$ and $\varepsilon:=\delta^2$. Consider the open set $W(x,y;\varepsilon)$ in $\tau_K$. Let $z\in W(x,y;\varepsilon)$; then
$$
\begin{equation*}
|\langle\overrightarrow{xz},\overrightarrow{xy}\rangle|= d(x,z)d(x,y)|\cos\theta|<\varepsilon,
\end{equation*}
\notag
$$
where $\theta$ is the comparison angle at vertex $\overline{x}$ in the Euclidean comparison triangle $\Delta(\overline{x}, \overline{y}, \overline{z})$ determined by the edge lengths of the triangle $\Delta(x,y,z)$. Suppose that $d(x,P_{\gamma}z)\geqslant \delta$. Without loss of generality assume that $P_{\gamma}z=y$. Since $\gamma$ is a closed convex set, by a property of projections the Alexandrov angle $\angle_y([y,x],[y,z])$ at $y$ between the geodesic segments $[y,x]$ and $[y,z]$ is at least $\pi/2$. Because of nonpositive curvature the comparison angle $\angle_{\overline{y}}([\overline{y},\overline{x}],[\overline{y},\overline{z}])$ is greater than or equal to $\pi/2$. Then the projection of $\overline{z}$ onto the line $\overline{\gamma}$ that extends beyond the segment $[\overline{x},\overline{y}]$ lies outside this segment, that is, $|\overline{x}P_{\overline{\gamma}}\overline{z}| \geqslant|\overline{x}\,\overline{y}|=\delta$. On the other hand we have $|\overline{x}P_{\overline{\gamma}}\overline{z}| =|\overline{x}\,\overline{z}|\cos\theta=d(x,z)\cos\theta$, which implies that
$$
\begin{equation*}
|\langle\overrightarrow{xz},\overrightarrow{xy}\rangle|= d(x,z)d(x,y)\cos\theta=|\overline{x}\,\overline{z}| \,|\overline{x}\,\overline{y}| \cos\theta=|\overline{x}P_{\overline{\gamma}}\overline{z}| \,|\overline{x}\,\overline{y}|\geqslant \delta^2=\varepsilon.
\end{equation*}
\notag
$$
This gives a contradiction, which proves (i). Theorem 3.8 and (i) imply (ii).
The theorem is proved. The above theorem ensures that our weak topology $\tau_w$ is coarser than the Kakavandi topology, regardless of whether or not the underlying space is weakly proper. The finer the topology, the fewer compactness results can be expected. Indeed, we are not aware of any compactness results for bounded closed convex sets in the Kakavandi topology. Note that Example 4.7 in [13] implies that $\tau_K$ can be strictly finer than $\tau_w$. 4.2. The Monod weak topologyMonod [20] proposed a topology on a Hadamard space $(H,d)$ which we denote by $\tau_{\mathrm{co}}$. The topology $\tau_{\mathrm{co}}$ is the weakest topology on $(H,d)$ such that any closed convex set is $\tau_{\mathrm{co}}$-closed. The Monod topology was studied in detail by Kell [14], who referred to it as the co-convex topology. The next result relates the topologies discussed so far. Proposition 4.1. The following chain of inclusions holds: $\tau_{\mathrm{co}}\subseteq\tau_w\subseteq\tau_K$. All three topologies coincide with the usual weak topology whenever $(H,d)$ is a Hilbert space. Proof. Let $(H,d)$ be a Hadamard space. Theorem 3.2 implies that a convex set is $\tau_w$-closed if and only if it is closed. Hence $\tau_{\mathrm{co}}\subseteq \tau_w$. The last statement of the theorem follows from the fact that Hilbert spaces are weakly proper, combined with the fact that $\tau_{\mathrm{co}}$ and $\tau_K$ coincide with the usual weak topology on any Hilbert space; see [20], Example 18. It is known that if $K\subseteq H$ is compact, then the restrictions of $\tau_{\mathrm{co}}$ and $\tau_s$ (the metric topology) to $K$ coincide (see [20], Lemma 17). Hence, in view of Proposition 4.1 the restrictions of all three weak topologies to a compact subset $K$ of a Hadamard space $H$ coincide with the strong topology. An important property of the Monod topology is that any closed convex and bounded set is $\tau_{\mathrm{co}}$-compact: see [20], Theorem 14. However, it turns out that, in general, a Hadamard space is not Hausdorff with respect to $\tau_{\mathrm{co}}$. For example, the Euclidean cone of an infinite-dimensional Hilbert space is not Hausdorff when equipped with $\tau_{\mathrm{co}}$ (see [14], Example 3.6). 4.3. Geodesically monotone operatorsA continuous operator $T\colon H\to H$ is said to be geodesically monotone if for all $x_0,x_1\in H$ the real-valued function $\varphi\colon [0,1]\to\mathbb{R}_+$ defined by $\varphi(\alpha;x_0,x_1):=d(Tx_0,Tx_{\alpha})$ is monotone in $\alpha$, where $x_{\alpha}:=(1-\alpha) x_0\oplus \alpha x_1$ is the convex combination along the geodesic from $x_0$ to $x_1$. The next theorem provides a sufficient condition for an arbitrary Hadamard space to be Hausdorff in $\tau_{\mathrm{co}}$. Theorem 4.2. If the projection $P_{\gamma}$ is geodesically monotone for all geodesic segments $\gamma$, then $(H,\tau_{\mathrm{co}})$ is Hausdorff. Proof. Let $x,y\in H$ be two distinct points and $\gamma\colon [0,1]\to H$ be a geodesic such that $\gamma(0)=x,\gamma(1)=y$. Let $l>0$ denote the length of $\gamma$. For some fixed number $0<\varepsilon<l$ let $C(x,\varepsilon):=\{z\in H\colon d(x,P_{\gamma}z)\leqslant \varepsilon\}$. We claim that $C(x,\varepsilon)$ is a closed convex set. Closedness follows immediately since $P_{\gamma}$ is nonexpansive and therefore continuous. Let $z_0,z_1\in C(x,\varepsilon)$ be two distinct elements. Let $z_{\alpha}:=(1-\alpha) z_0\oplus \alpha z_1$ for some $\alpha\in(0,1)$. By assumption $P_{\gamma}$ is a geodesically monotone operator; thus, $d(P_{\gamma}z_{0},P_{\gamma}z_{\alpha})$ is monotone in $\alpha$, implying that $P_{\gamma}z_{\alpha}\in[P_{\gamma}z_0,P_{\gamma}z_1]$. The estimate $d(x,P_{\gamma}z_{\alpha})\leqslant\max\{d(x,P_{\gamma}z_0), d(x,P_{\gamma}z_1)\}\leqslant\varepsilon$ implies that $z_{\alpha}\in C(x,\varepsilon)$. From the definition of $\tau_{\mathrm{co}}$ it follows that $C(x,\varepsilon)$ is $\tau_{\mathrm{co}}$-closed. Hence $H\setminus C(x,\varepsilon)$ is $\tau_{\mathrm{co}}$-open. By construction $y\in H\setminus C(x,\varepsilon)$. Using the same argument, it follows that $C(y,\varepsilon):=\{z\in H\colon d(y,P_{\gamma}z)\leqslant l-\varepsilon\}$ is $\tau_{\mathrm{co}}$-closed. Hence $H\setminus C(y,\varepsilon)$ is a $\tau_{\mathrm{co}}$-open set containing $x$. It is obvious from the construction that $(H\setminus C(x,\varepsilon))\cap (H\setminus C(y,\varepsilon))=\varnothing$. Therefore, $(H,\tau_{\mathrm{co}})$ is a Hausdorff space. The theorem is proved. The contrapositive of this statement, in combination with Example 3.6 in [14], shows that the projection $P_{\gamma}$ is not a geodesically monotone operator in a general Hadamard space. This is in contrast with projections onto geodesic segments in Hilbert spaces, which are always geodesically monotone. Notice that the converse of Theorem 4.2 is not true as Figure 1 shows. Another interesting implication from the example illustrated in Figure 1 relates to the so-called normal cone. Given a closed convex set $C\subseteq H$ and $p\in C$, we define the normal cone at $p\in C$ by $N(p,C):=\{x\in H\colon \angle_p([x,p],[p,y])\geqslant\pi/2\text{ for all }y\in C\}$. Figure 1 tells us that $N(p,C)$ is not convex. This is in contrast with a basic result in Hilbert spaces that $N(p,C)$ is always a closed convex set. Remark 4.1. The geodesic monotonicity of the metric projections onto geodesics coincides with the so-called ‘(nice) N-property’ discussed in [9]. 4.4. A discussionIn relation to weak convergence of bounded sequences in $\operatorname{CAT}(0)$-spaces, a topology $\tau$ on a $\operatorname{CAT}(0)$-space $(X,d)$ was recently introduced in [17] in the following way: a set $S\subseteq X$ is $\tau$-closed in $X$ if any bounded sequence $(x_n)_{n\in\mathbb N}$ in $S$ that weakly converges to $x\in X$ implies that $x\in S$. By construction $\tau$ is finer than $\tau_w$. This topology differs from $\tau_w$ in that $\tau$ characterizes only bounded sequences (nets). Now, concerning the example in [17], § 4, it can be proved that it is not Hausdorff in $\tau$, consequently, it cannot be Hausdorff in $\tau_w$ since the latter is a coarser topology. Then from Lemma 3.2 it follows that this example of an Hadamard space cannot be weakly proper. Subsequently, in [19] characterization and comparison theorems were obtained in terms of nets for the weak topologies discussed here, and their relations to the weak topology in [17] were considered. AcknowledgementsI thank my advisors for their unconditional help and an anonymous referee of my Ph.D. thesis for their corrections. I am also grateful to Prof. Igor Nikolaev for general discussions about $\operatorname{CAT}(0)$-spaces. Credit should be given to Dr. Ph. Miller, who noted that the assumption of weak properness in Theorem 3.2 would be superfluous. I would like to express my gratitude to an anonymous referee of this manuscript for their helpful suggestions. I am also grateful to A. R. Alimov, who translated this paper into Russian.
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