I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ?

$$ F(y) = \int K(x,y) f(x) \text{d}x $$ where $K(x,y) = \text{e}^{-ixy}$ for the case Fourier transform. The functions $F(y)$ and $f(x)$ can be seen as $\left<y|F\right>$ and $\left<x|f\right>$ respectively. In such a case the above integral equation can be rewritten as -

$$ \left< y|F \right> = \left<y|\mathbb{\hat I}|F\right> = \sum_x \left<y |x\right> \left<x |f\right> $$

So is $\left<y |x\right>$ one way of looking at the integral kernel for all general cases ? If not, I wish to understand how one can precisely look at integral kernels.

**EDIT 1:**
I also wish to know that can transforms like Laplace, Mellin etc. also be treated like that as Transformation matrix, also in which case it might not be unitary matrix in all cases, but rather just a map from one inner product space to another.