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Let Y\, be a closed subspace of a normed space (X,||\cdot ||_{2})\,. Define ||\cdot ||_{0}\, on the quotient space X/Y\, by

||{\hat  {x}}||_{0}={\mathrm  (}inf)_{{x\in {\hat  {x}}}}||x||\,

for every {\hat  {x}}\in X/Y\,. Prove that ||\cdot ||_{0}\, is a norm.