# Proof of closeness of kernel functions under pointwise product

How can I prove that pointwise product of two kernel functions is a kernel function?

By point-wise product, I assume you mean that if $k_1(x,y), k_2(x,y)$ are both valid kernel functions, then their product

is also a valid kernel function.

Proving this property is rather straightforward when we invoke Mercer’s theorem. Since $k_1, k_2$ are valid kernels, we know (via Mercer) that they must admit an inner product representation. Let $a$ denote the feature vector of $k_1$ and $b$ denote the same for $k_2$.

So $a$ is a function that produces an $M$-dim vector, and $b$ produces an $N$-dim vector.

Next, we just write the product in terms of $a$ and $b$, and perform some regrouping.

where $c(z)$ is an $M \cdot N$ -dimensional vector, s.t. $c_{mn}(z) = a_m(z) b_n(z)$.

Now, because we can write $k_p(x,y)$ as an inner product using the feature map $c$, we know $k_p$ is a valid kernel (via Mercer’s theorem). That’s all there is to it.