An Eberlein-Šmulian type result for the weak^{∗} topology

Hendrik Vogt

Abstract. We present a result on relative weak^{∗} compactness in the dual of a Banach space X that allows a short proof of both the Eberlein-Šmulian theorem and Šmulian’s characterisation of weak compactness of closed convex subsets of X.

Mathematics Subject Classification (2000). 46A50.

A subset A of a Banach space X is called conditionally countably weakly compact if every sequence in A has a weak accumulation point in X. The difficult implication of the Eberlein-Šmulian theorem states that such a set is already relatively weakly compact. This implication was proved by W. Eberlein in [Ebe47]; an elementary proof was given by R. Whitley [Whi67]. H. Cohen observed in [Coh67] that Whitley’s argument can also be used to show that a relatively weakly compact subset A of X is sequentially dense in the weak closure of A.

The aim of the present note is twofold. On the one hand, we apply Whitley’s argument and a version of Cohen’s argument to obtain an Eberlein-Šmulian type result for the weak^{∗} topology of the dual X′ (Proposition 2), thus avoiding the use of the bidual X′′ in these arguments. On the other hand, we add a further equivalence to the Eberlein-Šmulian theorem (condition (iv) in Theorem 3) that allows us to obtain Šmulian’s characterisation of weak compactness of closed convex subsets of X as an immediate corollary. In fact, the implication (iv) ⇒ (ii) of Theorem 3 is implicit in the proof of Šmulian’s theorem in [DuSc88; Thm. V.6.2].

The following lemma was already used in [Whi67]; the short proof is included for the convenience of the reader.

1. Lemma. Let X be a Banach space, and let Z be a finite dimensional subspace of X′. Then there exists a finite subset F of B_{X} such that

||x′|| ⩽ 2 max_{x∈F} |x′(x)| (x′ ∈ Z).

Proof. The unit sphere S_{Z} of Z is compact, so there exist x′_{1}, …, x′_{n} ∈ S_{Z} such that S_{Z} ⊆ ⋃_{k=1}^{n} B(x′_{k}, \frac{1}{4}). Let now x_{1}, …, x_{n} ∈ B_{X} such that x′_{k}(x_{k}) ⩾ \frac{3}{4} for k = 1, …, n. Then for all x′ ∈ S_{Z} there exists k ∈ {1, …, n} such that

|x′(x_{k})| ⩾ x′_{k}(x_{k}) − |(x′_{k} − x′)(x_{k})| ⩾ \frac{3}{4} − \frac{1}{4} = \frac{1}{2}||x′||,

and the assertion follows with F = {x_{1}, …, x_{n}}. □

The following result in particular gives a condition for a subset A of a dual space X′ to be relatively weakly^{∗} compact in the norm-closed linear hull of A.

2. Proposition. Let X be a Banach space, and let A ⊆ X′. Assume that for every sequence (a′_{n}) in A one has

⋂_{n∈ℕ} \overline{conv}\{a′_{k}; k ⩾ n\} ≠ ∅.

Then A is bounded, the weak^{∗} closure cl_{w^{∗}}A of A is contained in \overline{conv}A, and A is sequentially dense in cl_{w^{∗}}A.

Proof. Assume without loss of generality that X is a real Banach space. If A is not bounded, then by the uniform boundedness principle there exist x ∈ X and a sequence (a′_{n}) in A such that a′_{n}(x) → ∞ as n → ∞. This contradicts the assumption that ⋂_{n∈ℕ} \overline{conv}\{a′_{k}; k ⩾ n\} ≠ ∅.

Let now x′_{0} ∈ cl_{w^{∗}}A. Then by the definition of the weak^{∗}-neighbourhoods of x′_{0} and by Lemma 1 we can recursively choose an increasing sequence (F_{n})_{n∈ℕ_{0}} of finite subsets of B_{X} and a sequence (a′_{n})_{n∈ℕ} in A such that F_{0} = ∅,

|(x′_{0} − a′_{n})(x)| ⩽ \frac{1}{n} (x ∈ F_{n−1})

and

||x′|| ⩽ 2 max_{x∈F_{n}} |x′(x)| (x′ ∈ lin{x′_{0}, a′_{1}, …, a′_{n}}).

Let M := ⋃_{n∈ℕ} F_{n} and Y := lin({x′_{0}} ∪ {a′_{n}; n ∈ ℕ}). Then

||x′|| ⩽ 2 sup_{x∈M} |x′(x)|

holds for all x′ ∈ Y and hence for all x′ ∈ \overline{Y}.

By the assumption there exists y′ ∈ ⋂_{n∈ℕ} \overline{conv}\{a′_{k}; k ⩾ n\}. Then y′ ∈ \overline{conv}A, so the second assertion is proved if we show that y′ = x′_{0}. Let x ∈ M. Then there exists n_{0} ∈ ℕ such that x ∈ F_{n_{0}}. For n > n_{0} we have x ∈ F_{n−1} and hence |(x′_{0} − a′_{n})(x)| ⩽ \frac{1}{n}, so that |(x′_{0} − x′)(x)| ⩽ \frac{1}{n} for all x′ ∈ \overline{conv}\{a′_{k}; k ⩾ n\}. It follows that (x′_{0} − y′)(x) = 0, and as x′_{0} − y′ ∈ \overline{Y} we conclude that

||x′_{0} − y′|| ⩽ 2 sup_{x∈M} |(x′_{0} − y′)(x)| = 0.

Note that above we have shown that ⋂_{n∈ℕ} \overline{conv}\{a′_{k}; k ⩾ n\} = \{x′_{0}\}. In order to prove that A is sequentially dense in cl_{w^{∗}}A, we now show that in fact a′_{n} → x′_{0} weakly^{∗}. By the assumption we obtain for any subsequence (a′_{n_{j}}) of (a′_{n}) that

∅ ≠ ⋂_{k∈ℕ} \overline{conv}\{a′_{n_{j}}; j ⩾ k\} ⊆ ⋂_{k∈ℕ} \overline{conv}\{a′_{n}; n ⩾ k\} = \{x′_{0}\}

An Eberlein-Šmulian type result for the weak^{∗} topology 3

and hence x′_{0} ∈ \overline{conv}\{a′_{n_{j}}; j ∈ ℕ\}. Given x ∈ X, we infer that

x′_{0}(x) ∈ \overline{conv}\{a′_{n_{j}}(x); j ∈ ℕ\}

for any subsequence (a′_{n_{j}}(x)) of (a′_{n}(x)). From this property of the sequence (a′_{n}(x)) (of real numbers) one easily deduces that a′_{n}(x) → x′_{0}(x). □

With the above, the proof of the difficult implication of the Eberlein-Šmulian theorem is easy. We change perspective and apply Proposition 2 to the dual of a Banach space.

3. Theorem. (Eberlein-Šmulian) Let A be a subset of a Banach space X. Then the following are equivalent.

(i) A is relatively weakly compact; (ii) A is conditionally sequentially weakly compact; (iii) A is conditionally countably weakly compact; (iv) for every sequence (a_{n}) in A one has

⋂_{n∈ℕ} \overline{conv}\{a_{k}; k ⩾ n\} ≠ ∅.

Moreover, if one of the above conditions holds then the weak closure of A is equal to the weak sequential closure of A.

Proof. (i) ⇒ (ii). Let (a_{n}) be a sequence in A. Then X_{0} := \overline{lin}\{a_{n}; n ∈ ℕ\} is weakly closed, hence A_{0} := A ∩ X_{0} is a relatively weakly compact subset of X_{0}. Since X_{0} is separable, we thus obtain that the weak topology on A_{0} is metrisable. It follows that (a_{n}) has a subsequence that converges weakly in X_{0} and hence also weakly in X.

(ii) ⇒ (iii) is trivial.

(iii) ⇒ (iv) follows from the fact that the weak accumulation points of a sequence (a_{n}) in A are contained in ⋂_{n∈ℕ} \overline{conv}\{a_{k}; k ⩾ n\}.

(iv) ⇒ (i). By Proposition 2 we obtain that the weak^{∗} closure of A in X′′ is a bounded subset of \overline{conv}A ⊆ X. By the Banach-Alaoglu theorem this implies that A is relatively weakly compact. The proposition also yields equality of the weak closure of A and the weak sequential closure of A. □

As an immediate consequence of implication (iv) ⇒ (i) of the above theorem we obtain the following result.

4. Theorem. (Šmulian, [Smu39]) Let A be a closed convex subset of a Banach space X. Then A is weakly compact if and only if every decreasing sequence of non-empty closed convex subsets of A has a non-empty intersection.

4 Hendrik Vogt

References

[Coh67] H. B. Cohen: Sequential denseness and the Eberlein-Šmulian theorem. Math. Ann. 172 (1967), 209–210.

[DuSc88] N. Dunford and J. Schwartz: Linear Operators. Part I: General Theory. John Wiley, New York, 1988.

[Ebe47] W. F. Eberlein: Weak compactness in Banach spaces. I. Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 51–53.

[Smu39] V. Šmulian: On the principle of inclusion in the space of the type (B). Mat. Sbornik N.S. 5 (1939), 317–328.

[Whi67] R. Whitley: An elementary proof of the Eberlein-Šmulian theorem. Math. Ann. 172 (1967), 116–118.

Hendrik Vogt Fachrichtung Mathematik Technische Universität Dresden D-01062 Dresden Germany

e-mail: hendrhugo@egon ik.vogt@tu-darmstadt resden.de