How to express $k(x,y)=e^{x^Ty}$ as $\langle\phi(x),\phi(y)\rangle$?

Question

I am trying to show that $k(x,y)=e^{x^Ty}$ can be expressed as $\langle\phi(x),\phi(y)\rangle$.

I tried writing $k(x,y)=\sum_{j=0}^{\infty} \frac{1}{j!}(x^Ty)^j$. Now I know $(x^Ty)^j$ can be expressed as $\langle\phi(x),\phi(y)\rangle$ since I can verify it for the case $j=1,2,3$; but I don't see the exact formula to represent $(x^Ty)^j$ so that I can "separate" $x$ and $y$. Any help?

Answer 1

The thing you want to prove is false

for an actual inner product, the inner product of the zero vector with itself should be zero.