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Best Approximation In Inner Product Spaces

I can still remember being surprised and delighted to discover, back in high school, that drawing the 'best' straight line to fit a set of data was not a matter of artistic skill, but a mathematically precise problem with a simple, elegant solution. This problem, widely known as least-squares regression, is an elementary example in the theory of best approximation in inner product spaces. The general setup is as follows. Let X be an inner product space, with the norm defined as usual in terms of the inner product as x = x, x 1/2. Given a subset K of X and a point x ∈ X, we seek a point y 0 ∈ K such that y 0 − x = min y∈K y − x. Such a point y 0 , if it exists, is called a best approximation to x from K. It turns out that many approximation problems can be reformulated in these terms. Here are some of the questions that arise. Given x and K, does a best approximation y 0 exist? If so, then is it unique? How does one recognize it? Are there useful algorithms for finding it? How does it vary as x varies? Can one readily compute the error of approximation y 0 −x, or, if not, then at least estimate it? To answer these questions, it is necessary to develop a certain number of functional analytic tools. For example, the problem of existence and uniqueness