I Fourier transforms in Hilbert Space

[sonnichs]

I am thinking about the simple Fourier transform: F( sin(nx) ) <--> iπ( δ(ω+n) - δ(ω-n) }
I believe I can represent most of the functions in classical physics, f(x), in a Hilbert space by expressing them using their Fourier components as a basis.
This leads to a spectrum which is represented by the dirac-delta functions. Each axis in the space represents "n" for the delta and its length is the multiplier of the Fourier element of that function.

I can define a reciprocal space for these delta functions, each axis for δ(ω+n) being sin(nx).

I thought that such a space would also exist in the same Hilbert space--but am I incorrect? The delta function does not seem to comply with the rules of elements in a Hilbert space--it is a distribution--not a function?

thanks-Fritz

 

[Gavran]

I thought that such a space would also exist in the same Hilbert space--but am I incorrect? The delta function does not seem to comply with the rules of elements in a Hilbert space--it is a distribution--not a function?

See https://en.wikipedia.org/wiki/Rigged_Hilbert_space.

 

[sonnichs]

Thank you Gavran. This explains my premise. -FS