[[Vector space]]
# Inner product space

An **inner product space** is a vector space $(V, \mathbb{K})$[^K] together with an operation
$$
\begin{align*}
\langle \cdot, \cdot \rangle : V \times V \to \mathbb{K} 
\end{align*}
$$
which for any $x, y, z \in V$ and $\lambda, \mu \in \mathbb{K}$ has the following properties: #m/def/anal/vec

1. **conjugate symmetry** $\langle x,y \rangle = \overline{\langle y, x \rangle}$ ^IP1
2. **linearity in the first argument**^[Alternatively in the second, see [[#^IP2a|below]]] $\langle \lambda x + \mu y, z \rangle = \lambda \langle x , z \rangle + \mu \langle y,z \rangle$ ^IP2
3. **positive-definiteness** $\langle x,x \rangle > 0 \iff x \neq \mathbf{0}$ ^IP3

  [^K]: Where either $\mathbb{K} = \mathbb{C}$ or $\mathbb{K} = \mathbb{R}$, in the latter case conjugate symmetry ([[#^IP1|1]]) is just symmetry

In some fields a [[bra-ket notation]] style inner product is more common, signaled by a `|` instead of `,`[^con], in which case the [[#^IP2|second axiom]] is

2. **linearity in the second argument** $\langle x | \lambda y + \mu z \rangle = \lambda \langle x | y \rangle + \mu \langle x | z \rangle$ ^IP2a

[^con]: See [[Vector notation in these notes]]

Every inner product induced a [[Normed vector space|norm]] $\|x\|^2 = \langle x,x \rangle$,
and norms have a corresponding unique inner product iff the [[Parallelogram law]] holds.

## Properties

1. **antilineärity in the other argument:**  $\braket{ \lambda x + \mu y | z } = \lambda^* \braket{ x | z } + \mu^* \braket{ y | z }$
2. **general [[Cauchy-Schwarz inequality]]:** $\abs{\braket{ x | y }}^2 \leq \|x\|^2 \|y\|^2$

### Further properties

- A change of basis is also a [[Change of inner product]]
- [[The inner product is continuous]]

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#state/tidy | #lang/en | #SemBr