Question

real analysis

hint

9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits an isometric x e co is the subspace off" consisting of all x ε 0 as i-+ ao. Prove the following four statements co, and IIA-llyli. Moreover, every Λ є (co)is obtained in this way. In brief, (co)'- . vector space isomorphism between them.) 5.9d Let Mk = {x E Co : xn = 0 for n = k+1,k+2, ). Let Sk be a countable dense subset of Mk. Then USk is a countable dense subset of co and of e Let Q {Xa : A є 2"). Then Q is an uncountable set in any two different members of Q have distance 1 and

Answer 1

take (1, 1, 1 χ aen termm Seues
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