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## Elements of functional analysis with Maple

Maple

There are a lot of Maple applications in calculus, but there is a bit  of Maple applications in functional analysis (for example, see http://en.wikipedia.org/wiki/Functional_analysis ) . I would like to bring forward such an application. The main idea of functional analysis lies in consideration of functions as  the points of functional spaces. In particular, the distance between two functions and  two sets of functions is defined. We consider  the space  of all the continuous functions  on the interval [-1,1] with the Chebyshev metrics  and the space  of  all the square-integrable  in the sense of Lebesgue functions on the same interval  with the metrics   . Using the Direct Search v.2 package (see http://www.mapleprimes.com/posts/101374-DirectSearch-Optimization-Package-Version-2 ), we find these distances  between the function exp(x) and the set of all the polynomials of degree less than or equal to 3:

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Let us plot pol and

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We see that pol and pol1 differ from the MacLaurin polynomial of order 3

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