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Suppose, we have an approximation result for finite matrices of all orders $n \times n$. When can we push such a result in the case of infinite matrices or kernels, maybe via possibly some ultralimit technique? For example, from the case of hermitian positive definite matrices, positive semidefinite matrices to the case of positive definite kernels.

Is there any meta-principle which governs such a situation? I would love to see if there are any examples or non-examples known.

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    $\begingroup$ (Classical) operator-norm limits of finite-rank operators are compact... definitely a proper subset of all bounded operators on an infinite-dimensional Hilbert space. Although I'm certainly not an expert in non-standard analysis, it's unclear to me what would happen to a/the non-standard extension of this idea... $\endgroup$ Commented Jul 28 at 0:17
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    $\begingroup$ @paulgarrett: One can indeed use ultraproducts of Banach spaces to "lift" certain finite dimensional constructions or arguments to the infinite-dimensional case. For instance, this is a standard technique in dilation theory of linear operators (first used for this purpose by Bill Johnson, if I remember correctly). One of my personal favourites is in spectral theory, where ultrapowers turn approximate eigenvalues of operators into eigenvalues. But this is not really an approximation result as required by the OP - it rather allows us to treat some situations "as if they were finite-dimensional". $\endgroup$ Commented Jul 29 at 0:05
  • $\begingroup$ @OP: I could certainly write a few things about the usage of Banach space ultraproducts in operator theory. But to make this an appropriate answer to your question, I think it would be important to be a bit more specific regarding what you consider an "approximation result". Could you give one or two concrete examples of approximation results that you have in mind? $\endgroup$ Commented Jul 29 at 0:08
  • $\begingroup$ @JochenGlueck, very interesting! Thanks for the fact/idea! :) $\endgroup$ Commented Jul 29 at 0:11
  • $\begingroup$ @JochenGluek I had the positive definite kernels in my mind. But it would be nice to see some Banach space results. $\endgroup$ Commented Jul 30 at 6:49

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