Fréchet derivative

http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative  the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from functions of one variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the […]

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Best approximation theorem

Theorem Let X be an inner product space with induced norm, and  a non-empty, complete convex subset. Then, for all , there exists a unique best approximation a0 to x in A. Proof Suppose x = 0 (if not the case, consider A − {x} instead) and let . There exists a sequence (an) in Asuch that . We now prove that (an) is a Cauchy sequence. By the parallelogram rule, we get . Since A is convex,  so . Hence  as  which implies  as . In other words, (an) is a Cauchy sequence. Since A is complete, […]

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Best approximation theorem

Best approximation theorem Theorem Let X be an inner product space with induced norm, and  a non-empty, complete convex subset. Then, for all , there exists a unique best approximation a0 to x in A. Proof Suppose x = 0 (if not the case, consider A − {x} instead) and let . There exists a sequence (an) in Asuch that . We now prove that (an) is a Cauchy sequence. By the parallelogram rule, we get . Since A is convex,  so . Hence  as  which implies  as . In other words, (an) is […]

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Fréchet derivative

http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative  the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from functions of one variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the […]

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Ramda for reactive streams

Take a small Observable example from xstream – it outputs two numbers and completes when a second stream finishes after 5 seconds. It uses Observables and shows filtering and mapping values through the stream: 12345678910111213141516171819 import xs from ‘xstream’// Tick every second incremental numbers,// only pass even numbers, then map them to their square,// and […]

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