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### A Study on Best Approximation Elements in Normed Spaces

*Sahar Mohamed Ali Abo Bakr*

#### Abstract

In this paper I show that in a totally bounded metric space every sequence has a Cauchy subsequence, give an example of a totally bounded subset of a metric space which is not sequentially compact, and in a normed space X, I prove that, If W is the open unit ball of X, then any element x outside W has no best approximation from W and give the general formula of the distant from x to W, if W is the closed unit ball of X, then every x outside or on the boundary of W has best approximation and the set of best approximation elements is a subset of the unit sphere(boundary of W), I also determine the general formula for best approximation elements from the unit ball, sphere, and the closed unit ball of the given normed space in general. On the other side, away from the parallelogram identity assumption and the compactness assumption reduced assumptions are given for getting best approximation elements and improve some results in previous research papers. I supply the work with several examples.

#### Keywords

Normed and metric spaces, best approximation elements, compactness, totally boundedness, open and closed unit balls

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