[mlevental]

equation 1 (in formal definition). the basis is e^(-st). if you don't know how that's a basis you need to read a little bit about functional analysis but just look at the integral as a continuous sum and f(t) as the basis coefficients and e^(-st) starts to look like a vector space basis (hilbert space) basis.

[MauiWarrior]

I am afraid it is a bit over my head. Would you be able to point me to the source?

[peterhj]

Not the most explanatory sources but function spaces essentially are vector spaces:

https://en.wikipedia.org/wiki/Vector_space#Function_spaces

http://mathworld.wolfram.com/HilbertSpace.html

[mlevental]

https://www.youtube.com/watch?v=7ICYxBuS2iw looks like a pretty elementary introduction