Suppose we define an $\mathbb{R}^m$ inner product space in which the inner product of $\mathbf{x}$ and $\mathbf{y}$ is $\exp\left(-\|\mathbf{x} - \mathbf{y}\|\right)$. In PCA and machine learning, we say that there are infintely many dimensions in this inner product space.
Prove that there exists no finite-dimensional vector-valued function $\Phi$ such that $$\exp\left(-\|\mathbf{x} - \mathbf{y}\|^2\right)\propto \Phi\left(\mathbf{x}\right) \left[\Phi\left(\mathbf{y}\right)\right]^\mathrm{T},$$
and suggest somehow that the above statement is valid in some infinite-dimensional space.
Edit 0
Okay, so it's not an inner product space because it doesn't meet bilinearity. But can you still prove the above statement? That is, that the Gaussian formula is not an inner product in any finite-dimensioned $\Phi : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (where $m$ can be large) space defined as $\Phi\left(\mathbf{x}\right)$ for each $\mathbf{x}$ in $\mathbb{R}^n$.