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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?

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  • $\begingroup$ 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). $\endgroup$ – air Sep 12 '20 at 11:19
  • $\begingroup$ 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 $\endgroup$ – christopher Sep 12 '20 at 11:21
  • $\begingroup$ a function of two arguments can't be defined by using the same argument twice $\endgroup$ – carlo Sep 12 '20 at 11:46
  • $\begingroup$ @carlo even if we consider other entity to be 0? $\endgroup$ – christopher Sep 12 '20 at 11:56
  • $\begingroup$ wait... $x'$ can be any vector? I thought it was $x$ translated $\endgroup$ – carlo Sep 12 '20 at 12:02
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$k(\cdot, \cdot)$, 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.

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