Parseval's Theorem

Inner Product Space

$$\text{Inner Product Space}\subset \text{Normed Vector Space}\subset \text{Metric Space}\subset \text{Topological Space}$$

where

1. Topological Spaces use open sets to measure what is close by $N_{\epsilon }(x)$.
2. Metric Spaces use distances to measure $d(x,y)=|x-y|$
3. Normed Vector Spaces $V$ have a norm $N:V\to {\mathbb{R}}_{\ge 0}$ where $N(x)$ is the “length” of $\vec{x}$. The distance function is $d(x,y)=N(x-y)$.
4. Inner Product Spaces use the inner product $V\times V\to \mathbb{R}$ or $V\times V\to \mathbb{C}$. Note that this is not the same inner product as used in Trigonometric Functions.

${\ell }^2$ Space

We have the ${\ell }^2$ space, the space of elements $(x_1,x_2,\dots )$ where $x_i\in \mathbb{C}$ such that

$$\sum _{i=1}^{\infty }|x_i{|}^2$$

converges. For example,

$$x=(1,1/2,1/3,1/4,\dots )\in {\ell }^2$$

is true since it is the sum of $1/n^2$, and then apply the P-Test. Then

$$y=(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{4},\dots )\notin {\ell }^2$$

by P-Test again.

$L^2$ Space

$L^2$ space is the space of integrable elements over $[-\pi ,\pi ]$ where

$$f\sim g\text{ if }{\int }_{-\pi }^{\pi }|f-g{|}^{2}dx=0$$

We use the same inner product from Definition (Inner Product).

Parseval’s Theorem

$L^2([-\pi ,\pi ])$ is isomorphic to ${\ell }^2$ as an inner product space. Both are examples of Hilbert Spaces, which are complete inner product spaces. The formula is:

$${\int }_{-\pi }^{\pi }|f{|}^{2}dx=2\pi \sum _{k=-\infty }^{\infty }|c_k{|}^2$$

for $c_k\in {\ell }^2$. This shows that

$$⟨f,f⟩=2\pi ⟨\{c_k\},\{c_k\}⟩$$

For the engineers, this allows us to move from the time/space domain to the frequency domain.