Finite dimensional banach space

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1. Show that in any finite dimensional Banach space (Rk with any norm), for any closed, convex set C and any point x not in C , there is at least one nearest point y in C ; in other words, ly - x l= infz∈C lz - x l.

2. Give an example of a closed, convex set C in a Banach space S and an x ∈ S which has no nearest point in C . Hint: Let S be the space £1 of absolutely summable sequences with norm l{xn }l1 := },n |xn |. Let C := {{t j (1 + j )/j } j ≥1: t j ≥ 0, }, j t j = 1} and x = 0.

Reference no: EM13990821

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