Because of such relationships

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1. Give an example of a Banach space X , a closed convex set C in X, and a point u ∈ X which does not have a unique nearest point in C .
Hint: Let X = R2 with the norm l(x, y)l := max(|x |, |y|).

2. Let (X, l·l) be a real normed space, E a linear subspace, and h ∈ E t. Give a proof that h can be extended to be a member of X t (the Hahn-Banach the- orem, 6.1.4) based on Theorem 6.2.11. Hint: Let U := {x ∈ X : lx l 1}. (Because of such relationships, separation theorems for convex sets are sometimes called "geometric forms" of the Hahn-Banach theorem.)

Reference no: EM13990819

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