Math 171: Problem Set 7 Solutions
نویسنده
چکیده
And hence d(x, 0) ≥ lim inf d(0, xn)− lim sup d(xn, x) = 1 Thus x ∈ S. Hence S is closed. Clearly S is bounded since it is contained in B2(0). Define en = (0, 0, . . . , 0, 1, 0, . . . ) be the sequence with all 0’s except a 1 in the nth place. Notice that en ∈ S (for `, `∞, c0). However, notice also that since, in every case, d(en, em) ≥ 1 whenever n 6= m, then the sequence en has no convergent subsequence in `, `∞, and c0. Hence, by Theorem 43.5, S is not compact.
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