>>13898660suppose you have the complex inner product on L2
which naturally induces the norm
now consider the (dual) function
this function is not in L2, but it has some interesting properties w.r.t the inner product
which acts similar to how an inner product is used when doing a basis expansion. in this case, however, there are an uncountable number of dimensions. so
this is very similar to expanding geometric vectors w.r.t an orthonormal basis such that
and
but a little more nuanced because of the infinite dimensions.
note, this interpretation helps you keep straight when to conjugate and when not to. also, it helps motivate the 1/sqrt(2 pi) factor needed to make the dirac delta have a factor of 1. also also, it helps when you need to use any one of 16 different possible conventions when using fourier analysis (i.e. f vs omega, unitary vs not unitary, positive vs negative phase, and i'm blanking on the 4th variation atm but i usually can think of one more)
