Exercise 12.4: Uniqueness of Kernel
Say there is another psd kernel \(\tilde{k}\) which satisfies the reproducing property: \(\forall x \, , \tilde{k}(\cdot, x) \in \mathbb{H}\), and \(f(x) = \langle f , \tilde{k}(\cdot, x) \rangle\) for all \(x\) and \(f \in \mathbb{H}\). Then by applying the reproducing property repeatedly \begin{align} k(x, x’) = \langle k(\cdot, x) , k(\cdot, x’) \rangle = \langle k(\cdot, x) , \tilde{k}(\cdot, x’) \rangle = \langle \tilde{k}(\cdot, x) , \tilde{k}(\cdot, x’) \rangle = \tilde{k}(x, x’) \, , \end{align} for any two \(x, x'\), i.e., the two kernels are equal.