Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, suppose the kernel is $K(x,y)$, since $K(x,\cdot) \in \mathcal{K} $, $K$ is continuous seperately in $x$ and $y$. But it need not be continuous in two variable.


Please someone give me an example that Reproducing kernel need not be continuous in two variables.



Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $1=a\_0=a\_1>\ldots >a\_n>\ldots >0$, and let $e\_n$ be the "triangle" function that vanishes outside $(a\_{n+1},a\_{n-1})$, equals $1$ at $a\_n$, and interpolates linearly from $0$ to $1$ on $[a\_{n+1},a\_n]$ and from $1$ to $0$ on $[a\_n,a\_{n-1}]$.


Put $K(x,y)=\sum\_1^\infty e\_n(x)e\_n(y)$. This is well defined, vanishes for $x=0$ (or $y=0$), is separately continuous on $[0,1]\times [0,1]$, is positive definite, and $K(x,x)$ oscillates between $1$ (at $a\_n$) and $1/2$ (at $(a\_{n+1}+a\_n)/2)$ as $x\to 0$ (where $K(0,0)=0$).


The associated RKHS is the set of piecewise affine continuous functions $\sum\_1^\infty c\_ne\_n$ with $\sum c\_n^2<\infty$. The unit ball is not equicontinuous on $[0,1]$, i.e. it is not compact in $C([0,1])$, a necessary and sufficient condition for the continuity of $K$ on $[0,1]\times [0,1]$.


