Eberlein-Šmulian Theorem

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 $\struct {X, \norm {\, \cdot \,}_X}$.

Let $A \subseteq X$.


The following statements are equivalent:

$(1): \quad$ $\struct {A, w}$ is compact
$(2): \quad$ $\struct {A, w}$ is sequentially compact in itself
$(3): \quad$ $\struct {A, w}$ is countably compact.


Proof

Let $\cl_w$ be the closure taken in $\struct {X, w}$.

Let $\cl$ be the closure taken in $\struct {X, \norm {\, \cdot \,}_X}$.

$(1)$ implies $(3)$

This follows from Compact Space is Countably Compact.

$\Box$

$(2)$ implies $(3)$

This follows from Sequentially Compact Space is Countably Compact.

$\Box$

$(3)$ implies $(2)$

Suppose that $\struct {A, w}$ is countably compact.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $A$.

Let $S = \set {x_n : n \in \N}$.

We show that $\sequence {x_n}_{n \mathop \in \N}$ has a weakly convergent subsequence with a limit in $A$.

Since $\struct {A, w}$ is countably compact, $\sequence {x_n}_{n \mathop \in \N}$ has an accumulation point $x \in A$.

If $x_n = x$ for sufficiently large $n$, then by Constant Sequence in Topological Space Converges, we have that $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$.

Otherwise, $x_n \ne x$ infinitely often.

Passing to a subsequence, suppose that $x_n \ne x$ for all $n \in \N$.

By Accumulation Point of Sequence iff Cluster Point of Net, $x$ is a cluster point of $\sequence {x_n}_{n \mathop \in \N}$ when viewed as a net.

From Point is Cluster Point of Net iff Limit of Subnet, $x$ is the limit of a subnet of $\sequence {x_n}_{n \mathop \in \N}$.

Hence by Point in Set Closure iff Limit of Net, we have $x \in \map {\cl_w} S$.

Suppose that $x \in \map \cl S$.

Then by Point in Closure of Sequence in Metric Space is Term of Sequence or Subsequential Limit, a subsequence of $\sequence {x_n}_{n \mathop \in \N}$ converges in norm to $x$.

Else suppose that $x \not \in \map \cl S$.

By Topological Closure of Translation in Topological Vector Space is Translation of Topological Closure, we have ${\mathbf 0}_X \in \map {\cl_w} {S - x}$ and ${\mathbf 0}_X \not \in \map \cl {S - x}$.

Further, by Weakly Countably Compact Set in Normed Vector Space is Norm Bounded, $A$ and hence $S$ is bounded.

From Translation of Bounded Subset of Normed Vector Space is Bounded, we have that $S - x$ is bounded.

Hence by the Kadets-Pełczyński Criterion on the Existence of a Basic Sequence, $S - x$ contains a basic sequence $\sequence {y_n - x}_{n \mathop \in \N}$ with $y_n \in S$ for each $n \in \N$.

Since $\sequence {y_n}_{n \mathop \in \N}$ is in $S$, it has an accumulation point $y \in A$ in $\struct {X, w}$ by countable compactness.

That is, from Accumulation Point of Sequence iff Cluster Point of Net, $y$ is a cluster point of $\sequence {y_n}_{n \mathop \in \N}$.

Then from Point is Cluster Point of Net iff Limit of Subnet, there exists a directed set $\struct {\Lambda, \preceq}$ and a cofinal mapping $\phi : \Lambda \to \N$ such that $\family {y_{\map \phi \lambda} }_{\lambda \mathop \in \Lambda}$ is a subnet converging to $y$ in $\struct {X, w}$.

From Sum of Convergent Nets in Topological Vector Space is Convergent and Constant Sequence in Topological Space Converges, $\family {y_{\map \phi \lambda} - x}_{\lambda \mathop \in \Lambda}$ converges to $y - x$ in $\struct {X, w}$.

By Weak Cluster Point of Basic Sequence in Banach Space is Zero Vector, it follows that $y - x = {\mathbf 0}_X$ and so $y = x$.

Hence $x$ is the unique cluster point of $\sequence {y_n}_{n \mathop \in \N}$.

From Unique Accumulation Point of Sequence in Weakly Countably Compact Subset of Banach Space is Limit of Sequence, it follows that $\sequence {y_n}_{n \mathop \in \N}$ converges weakly to $x \in A$.

Write $y_n = x_{\map \mu n}$.

Recall that we assumed that we have $x_n \ne x$ for all $n \in \N$.

Hence from Weak Limit in Normed Vector Space is Unique we must have that $\sequence {\map \mu n}_{n \mathop \in \N}$ is unbounded.

We pick $n_1 = \map \mu 1$.

For $k \ge 2$, take $N_k > \max \set {N_1, \ldots, N_{k - 1} } + 1$ such that $\map \mu {N_k} > \max \set {\map \mu {N_1}, \ldots, \map \mu {N_k} }$.

Take $n_k = \map \mu {N_k}$.

Then $\sequence {x_{n_k} }_{k \mathop \in \N}$ is a subsequence of both $\sequence {y_n}_{n \mathop \in \N}$ and $\sequence {x_n}_{n \mathop \in \N}$.

Hence by Limit of Subsequence equals Limit of Sequence, $\sequence {x_{n_k} }_{k \mathop \in \N}$ converges weakly to $x \in A$.

Hence every sequence in $A$ has a weakly convergent subsequence with limit in $A$.

Hence $\struct {A, w}$ is sequentially compact in itself.

$\Box$

$(3)$ implies $(1)$

Suppose that $\struct {A, w}$ is countably compact.

Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $B_{X^{\ast \ast} }^-$ be the closed unit ball in $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$.

Let $w^\ast$ be the weak-$\ast$ topology on $X^{\ast \ast}$.

Let $\cl_{w^\ast}$ be the closure taken in $\struct {X^{\ast \ast}, w^\ast}$.

From Weakly Countably Compact Set in Normed Vector Space is Norm Bounded, we have that $A$ is bounded.

Let:

$\ds \delta = \sup_{x \mathop \in A} \norm x_X$

We have that $A \subseteq \delta B_X^-$.

Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.

From Evaluation Linear Transformation on Normed Vector Space is Linear Isometry, $\iota$ is a linear isometry.

Hence we have that:

$\iota B_X^- \subseteq B_{X^{\ast \ast} }^-$

Hence:

$\iota A \subseteq \delta B_{X^{\ast \ast} }^-$

From Topological Closure is Closed, $\map {\cl_{w^\ast} } {\iota A}$ is closed in $\struct {X^{\ast \ast}, w^\ast}$.

From the Banach-Alaoglu Theorem, $\struct {B_{X^{\ast \ast} }^-, w^\ast}$ is compact.

From Continuous Image of Compact Space is Compact, we have that $\struct {\delta B_{X^{\ast \ast} }^-, w^\ast}$ is compact.

From Compact Subspace of Hausdorff Space is Closed and Weak-* Topology is Hausdorff, $\delta B_{X^{\ast \ast} }^-$ is closed in $\struct {X^{\ast \ast}, w^\ast}$.

From Closed Subspace of Compact Space is Compact, $\struct {\map {\cl_{w^\ast} } {\iota A}, w^\ast}$ is compact.

We first show that $\struct {\map {\cl_w} A, w}$ is compact.

Aiming for a contradiction, suppose $\struct {\map {\cl_w} A, w}$ is not compact.

From Bounded Subset of Banach Space is Relatively Weakly Compact iff Weak-* Closure in Second Normed Dual is Contained in Embedding of Original Space, we therefore have that:

$\map {\cl_{w^\ast} } {\iota A} \not \subseteq \iota X$

Hence there exists:

$\phi \in \map {\cl_{w^\ast} } {\iota A} \setminus \iota X$

Since ${\mathbf 0}_{X^{\ast \ast} } = \iota {\mathbf 0}_X \in \iota X$, we have $\phi \ne {\mathbf 0}_{X^{\ast \ast} }$.

Hence there exists $f \in X^\ast$ such that $\map \phi f \ne 0$.

By replacing $f$ with:

$\ds \frac 2 {\map \phi f} f \in X^\ast$

we can assume that $\map \phi f > 1$.

Let:

$A_0 = \set {x \in A : \map \Re {\map f x} > 1}$

We open another contradiction assumption.

Aiming for a contradiction, suppose that $\struct {\map {\cl_w} {A_0}, w}$ is compact.

From Bounded Subset of Banach Space is Relatively Weakly Compact iff Weak-* Closure in Second Normed Dual is Contained in Embedding of Original Space, we then have that:

$\map {\cl_{w^\ast} } {\iota A_0} \subseteq \iota X$

We have:

$\map \phi f > 1$

and $\phi \in \map {\cl_{w^\ast} } {\iota A}$.

Hence there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {\iota x_\lambda}_{\lambda \mathop \in \Lambda}$, with $x_\lambda \in A$, converging to $\phi$.

From Characterization of Convergent Net in Weak-* Topology:

$\map {\iota x_\lambda} f \to \map \phi f$

That is:

$\map f {x_\lambda} \to \map \phi f$

From Complex-Valued Function is Continuous iff Real Part and Imaginary Part are Continuous and Characterization of Continuity in terms of Nets, we then have:

$\map \Re {\map f {x_\lambda} } \to \map \Re {\map \phi f}$

Since $\map \Re {\map \phi f} \in \openint 1 \infty$, there exists $\lambda_0 \in \Lambda$ such that for $\lambda \succeq \lambda_0$, we have:

$\map \Re {\map f {x_\lambda} } > 1$

From Tail of Convergent Net Converges, $\family {\iota x_\lambda}_{\lambda \succeq \lambda_0}$ is then a net in $\iota A_0$ converging to $\phi$ in $\struct {X^{\ast \ast}, w^\ast}$.

Hence $\phi \in \map {\cl_{w^\ast} } {\iota A_0} \setminus \iota X$.

This is a contradiction.

Hence we have that $\struct {\map {\cl_w} {A_0}, w}$ is not compact.

Further, take $x \in \map {\cl_w} {A_0}$.

From Point in Set Closure iff Limit of Net, there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ such that:

$\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges weakly to $x$.

Hence from Characterization of Convergent Net in Weak Topology, we have:

$\map f {x_\lambda} \to \map f x$

From Complex-Valued Function is Continuous iff Real Part and Imaginary Part are Continuous and Characterization of Continuity in terms of Nets, we then have:

$\map \Re {\map f {x_\lambda} } \to \map \Re {\map f x}$

We have $1 < \map \Re {\map f {x_\lambda} }$ for each $\lambda \in \Lambda$.

Hence from Inequality Rule for Real Convergent Nets, we have $\map \Re {\map f x} \ge 1$.

Hence $\map {\cl_w} {A_0} \subseteq \set {x \in X : \map \Re {\map f x} \ge 1}$.

In particular, ${\mathbf 0}_X \not \in \map {\cl_w} {A_0}$.

Further, $A_0 \subseteq A$, so $A_0$ is bounded.

From the Kadets-Pełczyński Criterion on the Existence of a Basic Sequence, $A_0$ then contains a basic sequence.

Since $\struct {A, w}$ is countably compact, $\sequence {x_n}_{n \mathop \in \N}$ has an accumulation point in $\struct {A, w}$.

From Accumulation Point of Sequence iff Cluster Point of Net, $\sequence {x_n}_{n \mathop \in \N}$ has a cluster point $x$ in $\struct {A, w}$.

From Weak Cluster Point of Basic Sequence in Banach Space is Zero Vector, it follows that $x = {\mathbf 0}_X$.

However, from Point is Cluster Point of Net iff Limit of Subnet and Point in Set Closure iff Limit of Net, we would then have ${\mathbf 0}_X \in \map {\cl_w} A$, a contradiction.

We conclude that $\struct {\map {\cl_w} A, w}$ is compact.

From Set is Closed iff Equals Topological Closure, it is enough to show that $A$ is closed in $\struct {X, w}$.

Let $x \in \map {\cl_w} A$.

We aim to show that $x \in A$.

From Weakly Compact Set in Banach Space is Angelic, $\struct {\map {\cl_w} A, w}$ is angelic.

Hence there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $A$ such that $x_n \to x$ in $\struct {X, w}$.

Since $\struct {A, w}$ is countably compact, $\sequence {x_n}_{n \mathop \in \N}$ has an accumulation point $y$ in $\struct {A, w}$.

From Accumulation Point of Sequence iff Cluster Point of Net, is a cluster point of $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {A, w}$.

From Point is Cluster Point of Net iff Limit of Subnet, $\sequence {x_n}_{n \mathop \in \N}$ has a subnet converging weakly to $y \in A$.

From Subnet of Convergent Net Converges to Same Limit, this subnet also converges weakly to $x$.

From Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff, $\struct {A, w}$ is Hausdorff.

From Characterization of Hausdorff Property in terms of Nets, we must have that $x = y$.

Hence $x \in A$.

Hence we have that $\map {\cl_w} A = A$.

Hence $\struct {A, w}$ is compact.

$\blacksquare$


Source of Name

This entry was named for William Frederick Eberlein and Witold Lwowitsch Schmulian.


Sources

  • 2016: Fernando Albiac and Nigel J. Kalton: Topics in Banach Space Theory (2nd ed.) ... (previous) ... (next): $1.6$: The Eberlein–Šmulian Theorem
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