If X is any countable, compact topological space, then consider the Banach space C(X). The dual space is the finite measures on X, but as any measure is countably additive, and X is countable, taking so some ordering , we get an isometric isomorphism between C(X)* and $\ell_1$. But not all such C(X) are isomorphic: you can find such X by taking a countable ordinal w and considering the closed interval [0,w]. This is all very nicely explained in the book "Topics in Banach Space Theory" by Albiac and Kalton.
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If X is any countable, compact topological space, then consider the Banach space C(X). The dual space is the finite measures on X, but as any measure is countably additive, and X is countable, taking so ordering, we get an isometric isomorphism between C(X)* and $\ell_1$. But not all such C(X) are isomorphic: you can find such X by taking a countable ordinal w and considering the closed interval [0,w]. This is all very nicely explained in the book "Topics in Banach Space Theory" by Albiac and Kalton. |
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