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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 $ℓ 2 ⁢ 0 n−1 ℓ 2 0 n 1$

$x= x 0 x 1 … x n - 1 x x 0 x 1 … x n - 1$ is a list of numbers (finite sequence). The inner product for our two spaces are as follows:

• Standard inner product $Rn n$:
$⟨x,y⟩=yT⁢x=∑i=0n−1 x i ⁢ y i x y y x i 0 n 1 x i y i$
1
• Standard 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 $0 n−1 0 n 1$ or periodic (with period $nn$) discrete time signals. $x 0 x 1 … x n - 1 x 0 x 1 … x n - 1$

$f∈ L 2 ⁢ a b f L 2 a b$ is a finite energy function on $a b a b$

Inner Product
$⟨f,g⟩=∫abf⁢t⁢g⁢t¯dt f g t a b f t g t$
3
Model for: continuous time signals on the interval $a b a b$ 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
$⟨x,y⟩=∑i=−∞∞x⁢i⁢y⁢i¯ x y i x i y i$
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
$⟨f,g⟩=∫−∞∞f⁢t⁢g⁢t¯dt f g t f t g t$
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.

• $L 2 ⁢ a b L 2 a b$ → Fourier series
• $ℓ 2 ⁢ 0 n−1 ℓ 2 0 n 1$ → 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: $L 1 ( ℝ ) L 1 ( ℝ )$, $∥f∥1=∫|f⁢t|dt 1 f t f t$. Try as you might, you can't find an inner product that induces this norm, i.e. a $⟨·,·⟩ · ·$ such that
$⟨f,f⟩=∫|f⁢t|2dt2=∥f∥12 f f t f t 2 2 1 f 2$
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.