By E. Poisson

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**Sample text**

Howe_ver, oftentimes a sequence can converge to a vector x* but x* is not an element of S. 3 Let S be the open interval (0,1) with the norm 11 x II = Ix I for xES. Let {xn} be the sequence of vectors (points) defined by Xn = ~ for each n? 1. , the set of all real numbers q such that q = min where m and n are integers). , T =R-Q). Let t be an element of T and let xn be the first n digits of the decimal expansion of t. Then each xn E S, and as n+oo xn+t f S. } n ~:' where xn = . L: I/j!. Then xn+x* = e as n-+eo (the term xn is the first n J=o terms of the Taylor series for eX at x = 1).

That is, the Best Approximation Method will determine the best approximation function (from a set of basis functions) in an J',2 norm in that the J',2 error in satisfying the given linear operator relationship and the boundary and initial conditions will be a minimum. 6. Linear Spaces The subject of linear spaces and linear operators is fundamental to the development of the theory utilized in methods of numerical approximation. , the real number field, R). Also let there be defined the operations of addition between every two elements of S and scalar multiplication between every element of S and every element of F such that for all x,y,z € S and a, i3 € F: 23 (i) x + y€ s.

2 is too general for practical purposes: the limit of the sequence, x*, must be known to prove {xn}+x* as n+ oo • 27 y x I 01 I I I I I I Fig. 2. ··. That is, for any n;::N(£), all of the remaining terms Xj of the sequence (j>n) lie in the closed £neighborhood N£ (x n). As £+0 the associated N( £) get larger, resulting in a parallel sequence of closed £-neighborhoods where each successive N £k+1 (xn) is contained in the previous N £k(xn) for £ k > £ k+1. This sequence of nested closed neighborhoods focuses about the limit vector x* asN(£)-+