Complex-valued lipschitz function

Reference no: EM13990771

1. Let (S, d) be a metric space, E ⊂ S, and f a complex-valued Lipschitz function defined on E , with | f (x ) - f (y)| ≤ Kd(x, y) for all x and y in E . Show that f can be extended to all of S with | f (u) - f (v)|≤ 21/2 Kd(u, v) for all u and v in S.

2. Let c0 denote the space of all sequences {xn } of real numbers which converge to 0 as n → ∞, with the norm 1{xn }1s := supn |xn |. Show that (c0, 1·1s ) is a Banach space and that its dual space is isometric to the space £1 of all summable sequences (L1 of the integers with counting measure). Show that c0 is not reflexive.

Reference no: EM13990771

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