In control and signal processing, it's well-known that the impulse response (forcing by $\delta(t)$) of a linear time-invariant system completely determines its behavior. In discrete time the argument proceeds by constructing a basis of the discrete delta functions
$$ \delta(n-m) = \begin{cases} 1, \ \ n = m\\ 0, \ \ n \neq m\\ \end{cases}, $$
and observing that an arbitrary signal $u(n)$ has the trivial reconstruction $$ u(n) = \sum_{m=-\infty}^\infty u(m)\delta(n-m). $$ This in turn allows the system transfer function $H$ (which maps $\ell^2$ into itself) to act on the basis rather than the coefficients. Since the output $y(n)$ is related to the input $u(n)$ via $y(n) = Hu(n)$, by defining impulse response $h(n) = H\delta(n)$ we obtain $$ y(n) = \sum_{m=-\infty}^\infty u(m)h(n-m). $$ Since $u(n)$ is arbitrary, all system specific information (e.g. BIBO stability) can be deduced from $h(n)$.
Speaking informally, the continuous time analog should be obtained by reducing $\delta(n)$ to $\delta(t)$, and the sum to an integral. But in this case we need deal with the notion of a continuous basis of $L^2$--and a basis of distributions at that! What is the state of formal work in this area and is there an easy entry/reference for someone with basic knowledge of analysis / topology / linear algebra and some knowledge of functional analysis?