ON WEAK^{∗} SEPARABLE SUBSETS OF

DUAL BANACH SPACES

Joe Howard

New Mexico Highlands University

Let X denote a Banach space, B_{X} its closed unit ball, and X′ be the Banach dual space of X. sX is the set of all weak^{∗} sequentially continuous functionals on X′, and tX the set of all f ∈ X′′ which are weak^{∗} continuous on all bounded weak^{∗}–separable subsets of X′. The sets sX and tX are closed subspaces of X′′. The relation JX ⊆ tX ⊆ sX ⊆ X′′ holds where J is the canonical map of X into X′′. It is well known that JX = sX is equivalent to X′ being a Mazur space (weak^{∗} sequentially continuous functionals on X′ are weak^{∗} continuous); that sX = X′′ is equivalent to X being a Grothendieck space (in X′, weak^{∗} convergent sequences are weakly convergent). By considering the set tX, we have the following.

\underline{Theorem 1}. The following are equivalent.

(a) tX = X′′.

(b) Every operator T: X → l_{∞} is weakly compact.

(c) X is a reflexive Banach space. \underline{Proof}. (a) ⇒ (b) Let f ∈ X′′. Then f is continuous on all bounded separable subsets

A′ of X′ with the weak^{∗} topology. Let T: X → l_{∞} be an operator. Since B′_{l_{∞}} is weak^{∗} separable, A′ = T′(B′_{l_{∞}}) is a bounded weak^{∗} separable set. Since f is weak^{∗} continuous on A′ by hypothesis, T′′(f) = f ∘ T′ is weak^{∗} continuous on B′_{l_{∞}}; thus, T′′(f) ∈ l_{∞}. Then T′′X′′ ⊆ Jl_{∞} and T is weakly compact.

(b) ⇒ (c) A bounded sequence (x′_{n}) in X′ defines an operator

T: X → l_{∞} by T x = (< x′_{n}, x >).

Since every operator is weakly compact, we obtain that every bounded sequence in X′ contains a weakly convergent subsequence. By Eberlein’s Theorem, this implies X′, hence, X is reflexive.

(c) ⇒ (a) is clear.

From this theorem, it follows that a Grothendieck space with the property that tX = sX must be reflexive. We now consider this property.

A set A′ ⊆ X′ is weak^{∗}–M compact if for every bounded sequence (x′_{n}) in A′, the weak^{∗} closure of (x′_{n}) has a weak^{∗} convergent subsequence. Note that if a set is weak^{∗} sequentially compact, then the set is weak^{∗}–M compact, but not conversely.

\underline{Theorem 2}. The following are equivalent.

(a) tX = sX.

(b) Every operator T: X → l_{∞} is such that T′: l′_{∞} → X′ maps bounded sets of l′_{∞} into weak^{∗}–M compact subsets of X′.

(c) B_{X′} is weak^{∗}–M compact. The proof of this theorem is very similar to the previous theorem. A limited set A of

X is a set such that for every weak^{∗}–null sequence (x′_{n}) in X, we have x′_{n}(x) → 0 uniformly for x ∈ A. If all limited subsets of E are relatively norm compact, then X is said to have the Gelfand–Phillips (GP) property. By using Theorem 2 and Corollary 2.3 of [1], it follows that if tX = sX, then X has the GP property.

\underline{Corollary 3}. If X′ is a Mazur space, then X has the GP property.

The converse of Corollary 3 is not true as the space C[0, ω_{1}] has the GP property (B_{C[0,ω_{1}]′} is weak^{∗} sequentially compact), but C[0, ω_{1}]′ is not a Mazur space (see [2]).

X is called realcompact if the weak topology on X is realcompact, that is, homeomophic to a closed subset of the product ℝ^{I} for some set I.

\underline{Theorem 4}. Suppose X is realcompact and tX = sX. Then X′ is a Mazur space. \underline{Proof}. By [3] (see Proposition 4.3 of [2]), we know that X is realcompact if and only

if JX coincides with the space rX of all f ∈ X′′ such that the restriction of f to every weak^{∗} separable closed subspace of X′ is continuous. Since each weak^{∗} separable closed subspace of X′ can be generated by the weak^{∗} closure of a bounded sequence (x′_{n}), then in each weak^{∗} separable subspace is a weak^{∗} convergent subsequence (since tX = sX), and conversely. Hence, JX = sX, so X′ is a Mazur space.

\underline{References}

1. L. Drewnowski, “On Banach Spaces with the Gelfand–Phillips Property,” Math. Z., 193 (1986), 405–411.

2. G. A. Edgar, “Measurability in a Banach Space, II,” Indiana University Mathematics Journal, 28 (1979), 559–579.

3. M. Valdivia, Topics in Locally Convex Space, North Holland. æ