By James G. Simmonds

During this textual content which steadily develops the instruments for formulating and manipulating the sphere equations of Continuum Mechanics, the maths of tensor research is brought in 4, well-separated phases, and the actual interpretation and alertness of vectors and tensors are under pressure all through. This new version includes extra routines. moreover, the writer has appended a bit on Differential Geometry

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Be a denumerable set, dense in H. Let e' and e" be two distinct given vectors of our orthonormalized system. Let us associate with each of these vectors an element of {a;fc} which is distant less than from its corresponding vector. , let II a*, - e' || < \ /2 , ||xk; - e" || < \ |/2. I. Approximation in Linear Normalized Spaces 26 If we show that k' =1= k ", then no more than a single vector of our orthonormalized system can correspond to each element of {xfc}, and our proposition will be proved.

I. Approximation in Linear Normalized Spaces 24 The previous section leads naturally to the following query: Given, in //, the linear independent vectors (1) i7i> 02> ••• 9 9n ; is it possible to find an equivalent orthogonal vector system < • m * 9 & n ? In other words, can the system (1) be orthogonalized ? It turns out that the orthogonalization is always possible. This can be done by a simple method due to E. Schmidt [10], which we will now explain. Let us put and recall that ||g1 1| =(= 0 by virtue of the' linear independence of the vectors (1).

N. Bernstein [13]. This proof is accomplished by constructing the polynomial =2 (T)**(1 - ) (n ^ 1) and then showing that lim B n (x) = f ( x ) uniformly throughout the interval [0, 1]. First of all, we have the following identities: ( 1) The last two identities are found by differentiating the relation K= V with respect to p. From the above identities, it follows that Multiplying (1) by f (x) and subtracting Bn (rr), we get m - iu * ) = 2 { / m - / ( ¿ ) } ( i ) ** = 2 + 2 • 20. Weierstrass’ First Theorem In the last expression, the summation in which k 1 n 31 runs through the values of k for yn and the summation in 2 " runs through the remaining values of k.