Math. Slovaca 37 (1987), 85–78

WEAK COMPACTNESS AND SUMMABILITY

WOJCIECH CHOJNACKI

Let T = (c_{nm})_{n,m∈ℕ} be a method of summability. In agreement with the terminology employed in [2], we will say that T is almost regular^{∗} if the following conditions are satisfied:

(i) lim_{n→∞} ∑_{m=1}^{∞} c_{nm} = c; (ii) lim_{n→∞} c_{nm} = c_{m} for every m ∈ ℕ; (iii) c ≠ ∑_{n=1}^{∞} c_{n}.

Here, of course, c and c_{n} (n ∈ ℕ) are finite, and the series are supposed to be convergent.

We will say that a subset of a Banach space has property 𝒮 if for every sequence in the subset there is an almost regular^{∗} summability method T such that the T - means of the sequence converge weakly.

In [2] D. Waterman established a theorem which can be formulated as saying that the unit ball of a Banach space having property 𝒮 is weakly compact. We shall prove the following generalization of this result.

Theorem. If a bounded subset of a Banach space has property 𝒮, then it is weakly relatively compact.

Proof. Suppose that a bounded subset A of a Banach space E has property 𝒮. Should not the weak closure of A be weakly compact, then, by a result of Kadec and Pełczyński [1], there is a basic sequence (x_{n})_{n∈ℕ} in A for which the origin is not a weak cluster point. By passing to a subsequence if necessary, we can assume that there exists x^{∗} in E^{∗}, the dual space of E, such that

∑_{n=1}^{∞} |x^{∗}(x_{n}) − 1| < ∞.

Of course, D = inf{∥x_{n}∥: n ∈ ℕ} > 0. Let C be a positive number such that, for every choice of n, m ∈ ℕ with n < m and scalars λ_{i} (1 ≤ i ≤ m), we have

∥ ∑_{i=1}^{n} λ_{i}x_{i}∥ ≤ C∥ ∑_{i=1}^{m} λ_{i}x_{i}∥.

Then

| ∑_{i=1}^{m} λ_{i}| ≤ | ∑_{i=1}^{m} λ_{i}(x^{∗}(x_{i}) − 1)| + | ∑_{i=1}^{m} λ_{i}x^{∗}(x_{i})|

≤ D^{−1} max{∥λ_{i}x_{i}∥: 1 ≤ i ≤ m} ∑_{i=1}^{m} |x^{∗}(x_{i}) − 1|

+ ∥x^{∗}∥∥ ∑_{i=1}^{m} λ_{i}x_{i}∥

≤ (2CD^{−1} ∑_{i=1}^{∞} |x^{∗}(x_{i}) − 1| + ∥x^{∗}∥)∥ ∑_{i=1}^{m} λ_{i}x_{i}∥.

This inequality jointly with the Hahn-Banach theorem shows that there is z^{∗} in E^{∗} such that

(1)  z^{∗}(x_{n}) = 1

for all n ∈ ℕ.

According to our hypothesis, there is an almost regular^{∗} summability method T = (c_{nm})_{n,m∈ℕ} and a point x in the closed linear span of {x_{n}: n ∈ ℕ} such that the T -means of (x_{n})_{n∈ℕ} converge weakly to x. Let (x_{n}^{∗})_{n∈ℕ} be a sequence in E^{∗} such that

x_{n}^{∗}(x_{m}) = \begin{cases}1 & if n = m, \\ 0 & if n ≠ m. \end{cases}

In view of (ii), for each n ∈ ℕ, the T -means of (x_{n}^{∗}(x_{m}))_{m∈ℕ} converge to c_{n}. Hence

x = ∑_{n=1}^{∞} c_{n}x_{n}

and further, in view of (1),

z^{∗}(x) = ∑_{n=1}^{∞} c_{n}.

On the other hand, by virtue of (i) and (1), the T -means of (z^{∗}(x_{n}))_{n∈ℕ} converge to c. This and the above equality imply

∑_{n=1}^{∞} c_{n} = c,

which contradicts (iii).

The proof is complete.

References

1. M. I. Kadec and A. Pełczyński, Basic sequences, bi-orthogonal systesm and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297–323 (Russian).

2. D. Waterman, Reflexivity and summability, II, Studia Math. 32 (1969), 61–63.

Instytut Matematyki, Uniwersytet Warszawski, Pałac Kultury i Nauki, IX p., 00-901 Warszawa, Poland