Common Hilbert Spaces Below we will look at the four most common Hilbert spaces that you will have to deal with when discussing and manipulating signals and systems.

n (reals scalars) and n (complex scalars), also called ℓ 2 0 n 1 x x 0 x 1 … x n - 1 is a list of numbers (finite sequence). The inner product for our two spaces are as follows: Inner product n : x y y x i 0 n 1 x i y i Inner product n : x y y x i 0 n 1 x i y i Model for: Discrete time signals on the interval 0 n 1 or periodic (with period n) discrete time signals. x 0 x 1 … x n - 1

f L 2 a b is a finite energy function on a b Inner Product f g t a b f t g t Model for: continuous time signals on the interval a b or periodic (with period T b a ) continuous time signals

x ℓ 2 is an infinite sequence of numbers that's square-summable Inner product x y i x i y i Model for: discrete time, non-periodic signals

f L 2 is a finite energy function on all of . Inner product f g t f t g t Model for: continuous time, non-periodic signals

Associated Fourier Analysis Each of these 4 Hilbert spaces has a type of Fourier analysis associated with it. L 2 a b → Fourier series ℓ 2 0 n 1 → Discrete Fourier Transform L 2 → Fourier Transform ℓ 2 → Discrete Time Fourier Transform But all 4 of these are based on the same principles (Hilbert space). Not all normed spaces are Hilbert spaces For example: L 1 ( ℝ ) , 1 f t f t . Try as you might, you can't find an inner product that induces this norm, i.e. a · · such that f f t f t 2 2 1 f 2 In fact, of all the L p spaces, L 2 is the only one that is a Hilbert space.

Hilbert spaces are by far the nicest. If you use or study orthonormal basis expansion then you will start to see why this is true.