## constituents

- \(V\) a vector space
- \((u,v)\), an
*ordered*pair of vectors in \(V\) (its not commutative!)

## requirements

We define \(\langle u, v \rangle \in \mathbb{F}\) as the inner product of \((u,v)\) **in that order!**. It carries the following properties:

**positivity**: \(\langle v, v\rangle \geq 0, \forall v \in V\)**definiteness**: \(\langle v, v\rangle = 0\) IFF \(v = 0\)**additivity in the first slot**: \(\langle u+v, w\rangle = \langle u, w \rangle + \langle v, w \rangle\)**homogeneity in the first slot**: \(\langle \lambda u, v \rangle = \lambda \langle u, v \rangle\)**conjugate symmetry**: \(\langle u,v \rangle = \overline{\langle v,u \rangle}\)

## additional information

### Inner Product Space

An Inner Product Space is a vector space with a well-defined inner product. For instance, \(\mathbb{F}^{n}\) has the canonical inner product named Euclidean Inner Product (see below, a.k.a. dot product for reals). The existence of such a well-defined inner product makes \(\mathbb{F}^{n}\) an Inner Product Space.

Rare Axler moment, instead of “well-defined”, he says we want a vector space with an inner product “lurking nearby”; james bond style.

### properties of inner product

- For a fixed \(u \in V\), the function takes \(v\) to \(\langle v,u \rangle\) is a Linear Map \(V \to \mathbb{F}\)
- \(\langle 0,u \rangle = 0\)
- \(\langle u,0 \rangle = 0\)
- \(\langle u,v+w \rangle = \langle u,v \rangle + \langle u,w \rangle\)
- \(\langle u,\lambda v \rangle = \bar{\lambda}\langle u,v \rangle\)

Proof:

- Inheriting the additivity and homogeneity of the definition of inner products
- Set \(u\) to be the fixed element for 1), set \(0\) to be the input, linear maps take \(0\) to \(0\)
- Apply conjugate symmetry to 2)
- Apply conjugate symmetry, inner product additivty, then conjugate back
- Apply conjugate symmetry, inner product homogeneity in the first slot, then conjugate back (of course leaving \(\lambda\) out conjugated)

### Euclidean Inner Product

For \(x,y \in \mathbb{F}^{n}\), one can define a pretty well-defined inner product by

\begin{equation} x \cdot y = x_1 \bar{y_{1}} + … + x_{n} \bar{y_{n}} \end{equation}

similar to dot product for the reals. This is called the Euclidean Inner Product and has the nice parallelity properties we saw.

### complex number shenanigans that motivate the inner product

…as both relevant and more general than the dot product, but also different in key areas.

First, review complex numbers from our discussion in chapter 4. The main problem here is this:

for \(z = (z_1, \dots, z_{n}) \in \mathbb{C}^{n}\), simply squaring each slot to take the norm may cause us to take a square root of a negative number (as each slot would then be \(a^{2}-b^{2}\) for a complex number). That’s no bueno because we want \(\|z\|\) to be real and non-negative.

This, therefore, suggests something similar for our inner product definition; to make sure that each slot end up being a real and non-negative number, we simply conjugate the second value:

\begin{equation} x \cdot y = x_1 \bar{y_{1}} + … + x_{n} \bar{y_{n}} \end{equation}

Also, note that this definition give us an important result: if we reverse \(x\) and \(y\), we would be conjugating the other element! And so, we have that:

\begin{equation} x \cdot y = \bar{{y \cdot x}} \end{equation}

derived by following the usual rules of complex conjugation. Note that none of these elementwisethings (the \(x_{n}y_{n}\) business) are actually in the definition of the inner product, as it is the rules of an Euclidean Inner Product.

### inner product of \(L\) periodic functions

For \(f,g : [0,L] \to \mathbb{R}\), which are L-periodic, we define:

\begin{equation} \langle f,g \rangle := \frac{1}{L} \int_{0}^{L} f(x) g(x) \dd{x} \end{equation}

Recall that L-periodic functions can be shifted without changing periodicity. But if for some reason you want to base it off of any two numbers with distance \(L\) in between:

\begin{equation} \langle f,g \rangle_{[a,b]} := \frac{1}{b-a} \int^{b}_{a} f(x) g(x) \dd{x} \end{equation}

The work of checking this is a well-formed inner product is left to absolutely nobody.