FA17

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Let Y\, be a closed subspace of a normed space X\,. Let x_{0}\in X\, but not in Y\,. Use the Hahn-Banach Theorem to prove that there exists a bounded linear functional {\hat  {f}}\, on X\, such that

(a) {\hat  {f}}(y)=0\, for every y\in Y\,

(b) ||{\hat  {f}}||=1\,

(c) {\hat  {f}}(x_{0})=d(x_{0},Y)\,