*6. Let (12d) be a metric space, and ﬁx a point a E Y. For each point p E Y, deﬁne afunction fp : Y —> R by Mm) = arm) — den, 00 a) Prove that fp E 0.50”). b) Show that ”fit, — fq||oo = d(p, q) for all p,q E Y (thus, the map (I) :p —> fp is anisometry of Y into Cb(Y)). c) Prove that 050’) is complete in the uniform metric. Let Z be the closure of (NY)in 05,07). Conclude that Z is complete. Thus, every metric space Y is isometric to a dense subset of a complete metric spaceZ. The space Z is called the completion of Y. (It can be shown that Z is unique upto isometry, but you do not need to prove this.)
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