# Proving that a function is not a kernel function

The function is defined as

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

Norm in Hilbert Spaces can be defined as $$||x||= \sqrt{x^Tx}$$. I am not sure about the feature map of this function that how will it be and I am positive that it does not exist. Moreover, kernel function depends both on $$x$$ and $$x'$$ but how should we formally prove that condition?

• Is there a typo in the title? The kernel function depends on both $x,x'$. Also I think it's always nice when the main question includes all details required to answer (instead of "frontloading" some details to the title). – air Sep 12 '20 at 11:19
• No its not a typo! yes agreed it does depends on both $x$ and $x'$ but how to formally show that!. Yeah i am updating the title thanks – christopher Sep 12 '20 at 11:21
• a function of two arguments can't be defined by using the same argument twice – carlo Sep 12 '20 at 11:46
• @carlo even if we consider other entity to be 0? – christopher Sep 12 '20 at 11:56
• wait... $x'$ can be any vector? I thought it was $x$ translated – carlo Sep 12 '20 at 12:02

$$k(\cdot, \cdot)$$, depending on the first term, may be any positive number, and is independent on the second term, hence kernel matrix $$K$$ can be any matrix having equal positive values whitin rows.
Take a matrix $$K$$ all equal to 0 except for the first row, and make it equal to 1. Also take a vector $$c$$ equal to 1 in all its value except for $$c_1=-1$$: $$c^T K c < 0$$ as long as the dimension of the form is greater than 2.
In conclusion, $$k$$ is not positive definite.