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Common Hilbert Spaces

Module by: Roy Ha, Justin Romberg

Summary: This module will give an overview of the most common Hilbert spaces and their basic properties.

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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.

ℝn n (reals scalars) and ℂn n (complex scalars), also called ℓ 2 0n−1 ℓ 2 0 n 1

x= x 0 x 1 … x 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 ℝn n :
    <x,y>=yTx=∑i=0n−1 x i y i x y y x i 0 n 1 x i y i (1)
  • Inner product ℂn n :
    <x,y>=yT¯x=∑i=0n−1 x i y i ¯ x y y x i 0 n 1 x i y i (2)

Model for: Discrete time signals on the interval 0n−1 0 n 1 or periodic (with period nn) discrete time signals. x 0 x 1 … x n - 1 x 0 x 1 … x n - 1

Figure 1
Figure 1 (fig1.png)

f∈ L 2 ab f L 2 a b is a finite energy function on ab a b

Inner Product

<f,g>=∫abftgt¯dt f g t a b f t g t (3)
Model for: continuous time signals on the interval ab a b or periodic (with period T=b−a T b a ) continuous time signals

x∈ ℓ 2 ℤ x ℓ 2 is an infinite sequence of numbers that's square-summable

Inner product

<x,y>=∑i=-∞∞xiyi¯ x y i x i y i (4)
Model for: discrete time, non-periodic signals

f∈ L 2 ℝ f L 2 is a finite energy function on all of ℝ.

Inner product

<f,g>=∫-∞∞ftgt¯dt f g t f t g t (5)
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 ab L 2 a b → Fourier series
  • ℓ 2 0n−1 ℓ 2 0 n 1 → Discrete Fourier Transform
  • L 2 ℝ L 2 → Fourier Transform
  • ℓ 2 ℤ ℓ 2 → Discrete Time Fourier Transform
But all 4 of these are based on the same principles (Hilbert space).

Important note:

Not all normed spaces are Hilbert spaces
For example: L 1 ( ℝ ) L 1 ( ℝ ) , ∥f∥1=∫|ft|dt 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>=∫|ft|2dt2=∥f∥12 f f t f t 2 2 1 f 2 (6)
In fact, of all the L p ℝ L p spaces, L 2 ℝ L 2 is the only one that is a Hilbert space.

Figure 2
Figure 2 (fig2.png)

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.

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