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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Does the set of positive definite kernels on some set X form a ring?

Given some non-empty set $X$, does the set of positive definite kernels on $K_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K_X$ then we wou …
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