Preliminaries

Best Approximations in Normed Spaces#

Consider the following problem. Let \(X\) be a normed vector space, and let \(d(x,y)\) be the metric that is induced from the norm. Then an abstract version of best approximation is

Given a subset (even a subspace) \(Y\) of \(X\) and a point \(x\in X\), is there an element \(y\in Y\) that is nearest to \(x\)? That is, can we find a vector \(y\in Y\) such that \(||x - y|| = \min_{z\in Y}||x-z||\)? If there is sch a best approximation to \(x\) from elements of \(Y\), is it unique?

In general \(Y\) needs to be closed, for otherwise points in the boundary of the set \(Y\) will not have nearest points. There needs to be further requirements on \(Y\) in order to ensure the existence of nearest points.