The reproducing property and its consequences

ProofAdvanced⏱ 30 minClassical MLrkhsreproducing propertyevaluation functional

Problem

In the RKHS $\Hcal_K$ , prove that

|f(x)| \le \|f\|_{\Hcal_K}\sqrt{K(x,x)} \quad\text{for all } f,x,

and deduce that norm convergence in $\Hcal_K$ implies pointwise convergence. Explain why this fails in $L^2$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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