To check if the kernel $K$ (feature map function) is a valid kernel or not, $K$ must satisfying Mercer's condition.
Mercer's condition: The kenel $K$, is a valid kernel if and only if $K$ is positive definite.
This satisifed in case of there's a matrix $c$ such that
$M$ = $c^T K c$, where $K$ is the Gram matrix (Kernel resultant matrix).
$st:$ $M \geq 0, for \,\, all \,\, real \,\ value \,\ c_i$
Or
$\sum_{i=1}^{n}\sum_{j=1}^{n} c_i k_i{_j} c_j \geq 0 $
$\geq 0$ $\Leftrightarrow$ positive semi-definite
before going to the intuition of why using Mercer's condition & intuitive proof, I would like to mention that, check the existence of the kernel $K$ and Mercer's condition has nothing to do with feature map itself, however it's a crucial for the convergence of the quadratic program such (SVM), or more generally the convexity.
Intuition
$K$ is a symmetric matrix then by the spectral theorem $K$ is a diagonalizable matrix (in other words, we can decompose it), so we can reformulate K by eigendecomposition
$K = VDV^T$
where $V$ is an orthogonal matrix and its column are the eigenvectors, and $D$ is a diagonal matrix with eigenvalues $λ_i{_j}$ on the diagonal.
If : $\exists $ $λ_i{_j}$ $for \,\, all \,\, i=j$ $such \ that \ λ_i{_j} < 0$
($\exists $) $\Leftrightarrow$ There's exist
Then the kernel $K$ not a valid kernel, beacuse of the negative eigenvalue means, there's a such point at which the Hessian is indefinite, in other words, the critical point is a saddle (the function is strongly convex-concave), then the primal problem has no solution, and even dual problem could be very expensive to compute (arbitrarily large).
by Sylvester's criterion, $K$ has negative eigenvalues if and only if it is not positive semi-definite.
Geometric intuition
The feature mapping by kenel function or the inner product of the features vector (row matrix) $x_i$ , $x_j$, $K=\langle \phi(x_i)\,, \phi(x_j)\rangle$ is same as the mesurement of the similarity of two functions by the definition of Hilbert space concept, visually, the kernel function is a congruent transformation, which is a transformation in an isometric space, if we have a negative eigenvalue, then transformation has occurred in the opposite direction in isometric space, in other words, the image reflecte across some axis.