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[Music] so [Music] and welcome back to data driven methods in dynamical systems and today i want to talk to you about an inner product space that me and my colleagues have developed for the analysis of dynamical systems specifically dynamical systems that give rise to densely defined label operators and this involves occupational kernels label operators or crewmen generators and reproducing kernel helper spaces so inner products are absolutely essential for everything dealing with machine learning and approximation theory and in particular given a collection of basis functions you can make a span of these basis functions and if it's a finite collection then this gives us a close-up space of an inner product space or hilbert space in general what we have is we have a function that is outside of the span of the spaces function so we want to find the weights that give us the best possible approximation to that outside function ing to this basis and the determination of these weigh

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