Theorem: Relativized Eberlein-Šmulian Theorem

Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
{{TFAE}}:
:$(1): \quad$ $\struct {A, w}$ is relatively compact in $\struct {X, w}$
:$(2): \quad$ $\struct {A, w}$ is sequentially compact in $\struct {X, w}$
:$(3): \quad$ $\struct {A, w}$ is relatively countably compact in $\struct {X, w}$.

Proof:
=== $(1)$ implies $(2)$ ===
Suppose that $\struct {A, w}$ is relatively compact.
Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $A$.
From Characterization of Separable Normed Vector Space:
:$Y = \map \cl {\map \span {\set {a_n : n \in \N} } }$ is separable.
Let $\norm {\, \cdot \,}_Y$ be the restriction of $\norm {\, \cdot \,}_X$ to $Y$.
Let $\struct {Y^\ast, \norm {\, \cdot \,}_{Y^\ast} }$ be the normed dual space of $\struct {Y, \norm {\, \cdot \,}_Y}$. 
Let:
:$S_{X^\ast} = \set {f \in X^\ast : \norm f_{X^\ast} = 1}$
:$S_{Y^\ast} = \set {f \in Y^\ast : \norm f_{Y^\ast} = 1}$
Hence from Normed Dual of Separable Normed Vector Space contains Normalized Sequence Separating Points, there exists a sequence $\sequence {f_n}_{n \mathop \in \N}$ in $S_{Y^\ast}$ that separate the points of $Y$.
By the Hahn-Banach Theorem, we can extend each $f_n$ to a linear functional in $S_{X^\ast}$.
From the definition of a relatively compact set, $\struct {\map {\cl_w} A, w}$ is compact.
From Compact Subspace of Topological Vector Space is von Neumann-Bounded, it follows that $\struct {\map {\cl_w} A, w}$ is von Neumann-bounded.
From Subset of Normed Vector Space is von Neumann-Bounded in Weak Topology iff Norm Bounded, $\map {\cl_w} A$ is bounded.
Hence $\sequence {a_n}_{n \mathop \in \N}$ is a bounded sequence.
From Fundamental Property of Norm on Bounded Linear Functional, we have:
:$\ds \cmod {\map {f_i} {a_n} } \le \norm {a_n}_X$ for each $i \in \N$.
Hence $\sequence {\map {f_i} {a_n} }_{n \mathop \in \N}$ is a bounded sequence for each $i \in \N$.
From the Bolzano-Weierstrass Theorem, $\sequence {\map {f_1} {a_n} }_{n \mathop \in \N}$ has a convergent subsequence $\sequence {\map {f_1} {a_{\map {\phi_1} n} } }_{n \mathop \in \N}$.
Suppose that, for $k \ge 2$ we have constructed $\sequence {\map {\phi_1} n}_{n \mathop \in \N}, \ldots, \sequence {\map {\phi_k} n}_{n \mathop \in \N}$ are sequences such that:
:$\sequence {\map {\phi_{j + 1} } n}_{n \mathop \in \N}$ is a subsequence of $\sequence {\map {\phi_j} n}_{n \mathop \in \N}$ for $1 \le j \le k - 1$
and:
:$\sequence {\map {f_j} {a_{\map {\phi_j} n} } }_{n \mathop \in \N}$ converges for $1 \le j \le k$.
By the Bolzano-Weierstrass Theorem, $\sequence {\map {f_{k + 1} } {a_{\map {\phi_k} n} } }_{n \mathop \in \N}$ has a convergent subsequence $\sequence {\map {f_{k + 1} } {a_{\map {\phi_{k + 1} } n} } }_{n \mathop \in \N}$.
Then the construction may continue.
Hence, for each $k \in \N$ we have a sequence $\sequence {\map {\phi_k} n}_{n \mathop \in \N}$ such that:
:$\sequence {\map {\phi_{j + 1} } n}_{n \mathop \in \N}$ is a subsequence of $\sequence {\map {\phi_j} n}_{n \mathop \in \N}$ for all $j \in \N$
and:
:$\sequence {\map {f_j} {a_{\map {\phi_j} n} } }_{n \mathop \in \N}$ converges for all $j \in \N$. 
Take $n_k = \map {\phi_k} k$.
Note that $\sequence {n_k}_{k \ge j}$ is a subsequence of $\sequence {\map {\phi_j} n}_{n \mathop \in \N}$.
Hence from Convergent Subsequence of Cauchy Sequence: Normed Vector Space, we have that:
:$\ds \sequence {\map {f_j} {a_{n_k} } }_{k \mathop \in \N}$ converges for all $j \in \N$.
Since a convergent subsequence of $\sequence {a_{n_k} }_{k \mathop \in \N}$ is also a convergent subsequence of $\sequence {a_n}_{n \mathop \in \N}$, we may replace $\sequence {a_n}_{n \mathop \in \N}$ with $\sequence {a_{n_k} }_{k \mathop \in \N}$ without loss of generlaity.
We show that $\sequence {a_n}_{n \mathop \in \N}$ has a unique cluster point.
Recall that $\struct {\map {\cl_w} A, w}$ is compact.
From {{Corollary|Characterization of Compactness in terms of Nets}}, $\sequence {a_n}_{n \mathop \in \N}$ has at least one cluster point in $\struct {X, w}$.
Let $a, b$ be cluster points of $\sequence {a_n}_{n \mathop \in \N}$ in $\struct {X, w}$.
From Point is Cluster Point of Net iff Limit of Subnet, there exists directed sets $\struct {\Lambda, \preceq}$ and $\struct {M, \sqsubseteq}$ and cofinal mappings $n_\bullet : \Lambda \to \N$ and $m_\bullet : M \to \N$ such that:
:$\family {a_{n_\lambda} }_{\lambda \in \Lambda}$ and $\family {a_{m_\mu} }_{\mu \in M}$ are subnets of $\sequence {a_n}_{n \mathop \in \N}$ such that:
::$\family {a_{n_\lambda} }_{\lambda \in \Lambda}$ converges to $a$ and $\family {a_{m_\mu} }_{\mu \in M}$ converges to $b$ in $\struct {X, w}$.
From Characterization of Convergent Net in Weak Topology, we have:
:$\map {f_j} {a_{n_\lambda} } \to \map {f_j} a$ for each $j \in \N$
and:
:$\map {f_j} {a_{m_\mu} } \to \map {f_j} b$ for each $j \in \N$.
Then for each $j \in \N$:
:$\family {\map {f_j} {a_{n_\lambda} } }_{\lambda \in \Lambda}$ is a subnet of $\sequence {\map {f_j} {a_n} }_{n \mathop \in \N}$
and:
:$\family {\map {f_j} {a_{m_\mu} } }_{\mu \in M}$ is a subnet of $\sequence {\map {f_j} {a_n} }_{n \mathop \in \N}$
From Subnet of Convergent Net Converges to Same Limit, we have that:
:$\map {f_j} {a_n} \to \map {f_j} a$
and:
:$\map {f_j} {a_n} \to \map {f_j} b$ 
for each $j \in \N$.
From Characterization of Hausdorff Property in terms of Nets, we have $\map {f_j} a = \map {f_j} b$ for each $j \in \N$.
Note that both $a$ and $b$ are weak limits of nets valued in $Y$.
Hence $a, b \in \map {\cl_w} Y$.
From Closed Vector Subspace in Normed Vector Space is Weakly Closed and Set is Closed iff Equals Topological Closure, we have $\map {\cl_w} Y = Y$.
Hence we have $a, b \in Y$.
Since $\sequence {f_j}_{j \mathop \in \N}$ separate the points of $Y$, we have $a = b$. 
Hence $\sequence {a_n}_{n \mathop \in \N}$ has a unique cluster point. 
Denote this cluster point by $a$.
We aim to show that $\sequence {a_n}_{n \mathop \in \N}$ converges weakly to $a$.
We will then have shown that every sequence in $A$ has a weakly convergent subsequence.
{{AimForCont}} that $\sequence {a_n}_{n \mathop \in \N}$ does not converges weakly to $a$.
Then there exists $g \in X^\ast$ such that $\map g {a_n}$ does not converge to $\map g a$.
From the Fundamental Property of Norm on Bounded Linear Functional, we have that:
:$\cmod {\map g {a_n} } \le \norm g_{X^\ast} \norm {a_n}_X$
Recalling that $\sequence {a_n}_{n \mathop \in \N}$ is bounded, we obtain that $\sequence {\map g {a_n} }_{n \mathop \in \N}$ is bounded.
Hence from the Bolzano-Weierstrass Theorem, $\sequence {\map g {a_n} }_{n \mathop \in \N}$ has a convergent subsequence $\sequence {\map g {a_{n_k} } }_{k \mathop \in \N}$.
From Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit, there exists a choice of $\sequence {n_k}_{k \mathop \in \N}$ such that:
:$\map g {a_{n_k} } \to L \ne \map g a$
Applying {{Corollary|Characterization of Compactness in terms of Nets}} to $\sequence {a_{n_k} }_{k \mathop \in \N}$ and $\map {\cl_w} A$ again, we obtain a cluster point $\alpha$ of $\sequence {a_{n_k} }_{k \mathop \in \N}$ in $\struct {X, w}$. 
Then there exists a directed set $\struct {\Lambda, \preceq}$ and cofinal mapping $\ell_\bullet : \Lambda \to \N$ such that:
:$\family {a_{\ell_\lambda} }_{\lambda \in \Lambda}$ is a subnet of $\sequence {a_{n_k} }_{k \mathop \in \N}$
and:
:$\family {a_{\ell_\lambda} }_{\lambda \in \Lambda}$ converges to $\alpha$ in $\struct {X, w}$.
From Characterization of Convergent Net in Weak Topology, we have that:
:$\map g {a_{\ell_\lambda} } \to \map g \alpha$
However, from Subnet of Convergent Net Converges to Same Limit, we have that:
:$\map g {a_{\ell_\lambda} } \to L$
From Characterization of Hausdorff Property in terms of Nets, we have that:
:$\map g \alpha = L \ne \map g a$
Hence we have $\alpha \ne a$.
Hence $\alpha$ is a cluster point of $\sequence {a_n}_{n \mathop \in \N}$ distinct from $a$.
This is a contradiction.
Hence $\sequence {a_n}_{n \mathop \in \N}$ converges weakly to $a$.
Hence we have shown that every sequence in $A$ has a weakly convergent subsequence.
Hence $\struct {A, w}$ is sequentially compact.
{{qed|lemma}}