Inner Products

A vector space, $V$, with an inner product is called an inner product space.

An inner product, $(\cdot, \cdot)$, is a functional that takes two vectors from the vector space $V$, defined over a field $\mathbb{F}^{n}$, as an input and outputs a scalar, $(\cdot, \cdot): \mathbb{F}^{n} \otimes \mathbb{F}^{n} \rightarrow \mathbb{F}^{1}$.

Inner Product Conditions¶

Let $\psi, ~\sigma ~\in~V$ and $a \in \mathbb{F}^{1}$.

To be an inner product the functional $(\cdot, \cdot)$ must obeys the following properties:

1. $(\psi, \psi) \geq 0$ and $(\psi, \psi) = 0$ if and only if $\psi=0$
   • Positive-definiteness
2. $(\psi,\omega) = (\omega, \psi)^{*}$
   • Conjugate symmetry
   • If $V$ is defined over $\mathbb{R}$ then $(\psi,\omega) = (\omega, \psi)$
3. $(\psi, a \omega) = a (\psi, \omega)$
   • Linearity in the second argument
   • If $V$ is defined over $\mathbb{C}$ then $(a \psi,\omega) = \overline{a}(\psi, \omega)$
4. $(\psi, \omega + \sigma) = (\psi,\omega) + (\psi, \sigma)$
   • Additivity

Cauchy–Schwarz inequality¶

The Cauchy–Schwarz inequality is an upper bound on the inner product between two vectors, in any vector space, in terms of the norms of the vectors, where the norm is defined via the inner product.

Let $\bm{x}, \bm{y}~\in~V$, where $V$ is a inner product space defined over $\mathbb{F}$ with inner product $(\cdot, \cdot) \rightarrow \mathbb{F}^{1}$, then

where $\vert \cdot \vert$ is the absolute value and