# Proving that a kernel function is kernel

Let's suppose we have a kernel function

$$k(x,x')=10$$

In order to prove that this a valid kernel function there are generally two conditions

• It is symmetric
• There exists a map $$\varphi:R^d \rightarrow H$$ called kernel feature map into some high dimensional feature space $$H$$ such that $$\forall x,x' \ in \ R^d :k(x,x') = \ <\varphi(x),\varphi(x')>$$.

How to formally approach to prove these two conditions for this kernel function?

If I get you correctly, you simply want $$k(x, x')$$ to be the constant function $$10$$? In that case
• It is symmetric because for all $$x, x'$$: $$k(x, x')=10=k(x',x)$$
• You can take $$H = \mathbb{R}$$ and $$\varphi(x) = \sqrt{10}$$. Then for any $$x, x'$$: $$\langle\varphi(x),\varphi(x')\rangle = \sqrt{10}^2 = 10 = k(x,x')$$