Exercise 15.1 - Reproducing property

We have known that the inner product of feature vectors can yield a semi-positive definite kernel matrix. This exercise introduces one method of deducing feature vectors from a legal kernel matrix. Denote $\kappa ({\mathbf{\text{𝐱}}}_1,\mathbf{\text{𝐱}})$ by $f(\mathbf{\text{𝐱}})$ and $\kappa ({\mathbf{\text{𝐱}}}_2,\mathbf{\text{𝐱}})$ by $g(\mathbf{\text{𝐱}})$. From the definition:

Since $\mathbf{\text{𝐱}}$ can be chosen arbitrarily, we have the properties hold(the one for $g$ is obtained similarly):

Therefore:

This completes the proof.