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Let $Y$ be a subspace of the normed vector space $X$ and let $x\in X$. Define $$d(x,Y)=\inf\{\lVert x+y \rVert,y\in Y\}$$

Prove that $d(x,Y)=\sup\{|f(x)|:f\in X^*,\lVert f\rVert \leq 1, f|_Y=0\}$

I have proven that $d(x,Y)\geq \sup\{|f(x)|:f\in X^*,\lVert f\rVert \leq 1, f|_Y=0\}$. How to prove that the equality can be achieved?

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    Do you mean $d(x,Y)$? – 2017-02-25
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    @user160738 yes, that's a typo – 2017-02-25

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Hint: Use Hahn-Banach. Suppose $x \notin Y$ (otherwise the problem is trivial). Define a linear functional $g$ on $Y \oplus {\rm Span \ } (x)$ by $g(y + \lambda x) = \lambda$ for any $y \in Y$ and any $\lambda \in \mathbb C$. What is the norm of $g$? Then use Hahn-Banach to extend $g$ to the whole of $X$ while preserving the norm - what does this tell you?