A Conceptual History of Space and Symmetry by Pietro Giuseppe Fré

Author:Pietro Giuseppe Fré
Language: eng
Format: epub
ISBN: 9783319980232
Publisher: Springer International Publishing

(f)The hint where to look for basis of functions is provided by Weierstrass theorem stating that a continuous bounded function can be approximated with arbitrary accuracy by a polynomial of sufficiently large degree. Since every element of can be seen as the limit of a succession of continuous functions, it follows that the polynomials and in particular the monomials provide a basis of functions. Typically such a basis is not orthonormal.

(g)The Gram–Schmidt orthonormalization algorithm utilized in finite dimensional vector spaces is iterative and inductive. Hence it can be easily extended to the infinite dimensional case. This shows that for one can find orthonormal bases composed of orthogonal polynomials.

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