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I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such that: $$T_\mathcal{K}(f)(x)=\int \mathcal{K}(x,y)f(y)dy$$

What I want to do is to show that if property $\phi$ holds for any gram matrix $K$ of $\mathcal{K}$ then it holds for $\mathcal K$ as well. A bad example would be $\phi(x):$"positive-defnitieness", since in this case it is defined to be like that. So the result that I am looking for concerns extending a property from finite dimensional matrix operations to infinite dimensional Kernel operator. I appreciate any guidance on ways to do this.

Now my idea is that in order to achieve this one might use Banach-Steinhaus theorem; it can be easily shown that the image of $f\sum\limits_{i=1}^{n}\delta_{x_i}$ under $T_\mathcal{K}$ will provide one with the result of multiplication of gram matrices of $\mathcal{K}$ with finite length vectors. This in turn enables one to use Banach-Steinhaus theorem since one can prove the property holds for matrix multiplications with finite vectors and therefore it holds for integrator operator. However the problem is Dirac delta function is not in the Hilbert space of say $L^2$ functions where $f$ belongs to and therefore talking about it being dense subspace seems nonsense. Is there any workaround to solve this problem?

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