By Gianni Gilardi
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Therefore I F (t) — U (i) sin t — V |~ cos t j ^ e , and consequently the trigonometric sum S (t) = U (t) sin t + V |— — tj cos t fulfills all our requirements. %%. The Separability of the Spaced. We are going to show that the space C (with respect to a finite interval) is separable. According to Weierstrass’ theorem, the aggregate of all polynomials is dense in C. If we now replace the (in general, complex) coefficients of any one of the polynomials by others sufficiently near these given numbers and having rational real and imaginary parts (these will be called briefly rational complex numbers), we shall obtain a polynomial which differs from the original by as little as we please.
X — liT-i)• As the polynomials A (x), B(x) have no common divisor, we can find polynomials
I IT. Orthogonalization of Vector Systems. Two vector systems 9 n 92* • • • > 9n 9i>92 f> - - , 9 k are said to be equivalent if they span one and the same subspace G. This means that every vector of the one system can be represented as a linear combination of the vectors of the other system. It is easy to see that the necessary and sufficient condition for this is 9Î = alkgl + a2kg2 + . . + ankgn with (k = 1, 2, . . , n) a12 . . aln a2x a22 *••a2n >. ann ani a7l2 •« It is clear that in the problems of approximation in which we are interested, a system can always be replaced by an equivalent system.
Analisi 1 by Gianni Gilardi