By Ivar Ekeland, Roger Témam
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Extra resources for Analyse convexe et problèmes variationnels (Etudes mathématiques)
Therefore, b + O, = b Comparing this to equation (2) and remembering that according to IV there exists only one solution of the equation b + x = b, we finally reach the equality 32 This now proves that in any field P there is a zero element, i. e. such an element 0 that for all a in P the equality a+O=a holds and therefore Condition V becomes completely meaningful. We already have three examples of fields - the field of rational numbers, that of real numbers, and that of complex numberswhile the sets of all integers and of positive real numbers do not constitute fields.
For instance, many different fields are contained within the fields of real numbers and of complex numbers; these are the so-called numerical fields. In addition some fields are larger than that of complex numbers. The elements of these fields are no longer called numbers, but the fields formed by them are used in mathematical research. Here is one example of such a field. Let us consider all possible polynomials f( x ) = aoxn + alx n-l + ... + an-IX + an with arbitrary complex coefficients and of arbitrary degrees; for instance, zero-degree polynomials will be represented by complex numbers themselves.
Therefore, since f(a) is positive 29 and j(b) negative, the second formula for the bound d must be used. 0215 ... 7784 ... 7769 ... 7784 ... 7785 It follows, therefore, that if we take for (X2 the arithmetic mean, i. e. 0008, equal to half the difference of these bounds. If the resulting accuracy is insufficient, we could once again apply the above method to the new bounds of the root (X2. However, this would require much more complicated calculations. Other methods of approximate solution of equations are more accurate.
Analyse convexe et problèmes variationnels (Etudes mathématiques) by Ivar Ekeland, Roger Témam