Reducing Kernel Hilbert Space: Reproducing property

Question

If the inner product between two functions is the $\int f(x)g(x)dx$ and it's equal to $\int f(x)g(x)dx = f(x)$, what conditions must $g(x)$ satisfy in order for this equality to hold?

Answer 1

The space in which $$<f,g> = \int f(x)g(x) \mathrm dx$$ is called the (real) $L^2$ space. In $L^2$ the representer of the evaluation functional is the dirac delta function, as $$ \int f(x) \delta(x-y) \mathrm d x = f(y)$$ however $\delta \notin L^2$, so $L^2$ is not a RKHS!