I'm trying to prove Corollary 5.4.6. from Albiac, Kalton - Topics in Banach Space Theory:
If $ K $ is a compact Hausdorff space then $ \mathcal{M}(K) $ has (DPP).
I want to use
The Dunford-Pettis Theorem. If $ \mu $ is a $ \sigma $-finite measure then $ L_1(\mu) $ has (DPP).
Definitions from the book:
Let $ X $ and $ Y $ be Banach spaces. A bounded linear operator $ T: X \to Y $ is a Dunford-Pettis operator if whenever $ W $ is weakly compact subset of $ X $ then $ T(W) $ is a norm-compact subset of $ Y $.
A Banach space $ X $ is said to have the Dunford-Pettis property (DPP) is every weakly compact operator $ T $ from $ X $ into a Banach Space $ Y $ is Dunford-Pettis.
My progress so far:
Let $ T: \mathcal{M}(K) \to Y $ be a weakly compact operator. The space $ \mathcal{M}(K) $ can be represented as an $ l_1 $-sum of spaces $ L_1(\mu_n) $, where $ \mu_n $ are probability measures on $ K $. We define $$ T_n: L_1(\mu_n) \to Y,\quad T_n(f) = T((0, \ldots, 0, f, 0, \ldots)) $$ (where $ f $ is at the $ n $-th position in the sequence $ (0, \ldots, 0, f, 0, \ldots) $).
Then for $ \mu = (f_1, f_2, \ldots) $ we have $$ T(\mu) = \sum\limits_{n = 1}^\infty T_n(f_n). $$ I guess the proof should go like this
- Because $ T $ is weakly compact, each $ T_n $ is weakly compact.
- By the Dunford-Pettis theorem, each $ T_n $ is Dunford-Pettis.
- From 2. it should follow that $ T $ is Dunford-Pettis.
My question
How do I prove the corollary stated above? Specifically, how do I prove 3.? Or should I try a completely different approach?