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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
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Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition
To answer your second question:
$H_D = \operatorname{span}(k_x : x \in D) $ is a finite dimensional (and hence closed) subspace of $H$. A function $f \in H$ is orthogonal to each $k_x, x \in D$ prec …