I can think of two differing viewpoints, and though many would prefer the latter due to its simplicity inherited from the Hilbert structure, I prefer the former for its connections to Sobolev spaces and PDE.
I. The Laplacian can be defined in the sense of distributions, since the Laplacian of an $L^2$ function is a distribution. In particular, for $u \in L^2(\Omega)$ define
$<\Delta u, \phi> := \int_\Omega u \Delta \phi\;dx$
for $\phi \in C^\infty_c(\Omega)$.
This would then say that in one dimension, for example, $\Delta |x| = 2\delta_0$, where I have used $\delta_0$ to denote the Dirac mass at zero, a distribution/measure.
II. For $u \in L^2(0,1)$, $sin(n\pi x)$ and $cos(n\pi x)$ form a basis and we can write
$u(x)= b_0 + \sum_n a_n sin(n\pi x) + b_n cos(n\pi x)$,
and we have $\sum_n a_n^2+b_n^2 <\infty$
Then we can define, formally, $\Delta u := \sum_n (n\pi)^2(-a_n sin(n\pi x) - b_n cos(n\pi x))$. In general, $\Delta u$ will not make sense pointwise, unless we know that $\sum_n n^4(a_n^2+b_n^2) <\infty$ (but as above, we can write $<\Delta u, \phi> = \int u \Delta \phi\;dx = \sum_n (n\pi)^2 (-a_n a_n^\prime-b_nb_n^\prime)$
where $a_n^\prime$ and $b_n^\prime$ are the Fourier coefficients of $\phi$. Now this makes sense for any $u \in L^2(0,1)$ if $\phi$ satisfies $\sum_n n^4((a_n^\prime)^2+(b^\prime_n)^2)<\infty$.
Of course, II can be done in higher dimensions - I have only chosen one dimension to illustrate with a simple example the basis functions. I do comment that I is more general, since it does not require the Hilbert structure.