# Inner Product

- operates on two vectors and produces a scalar
- notation $`\braket{x|y}`$
  - $`\bra{x}`$ is a [[bra]]
  - $`\ket{y}`$ is a [[ket]]
- follows laws
  1. $`\braket{0|y} = 0`$ and $`\braket{x|0} = 0`$
  2. $`\braket{x + y|z} = \braket{x|z} + \braket{y|z}`$ and $`\braket{x|y + z} = \braket{x|y} + \braket{x|z}`$
  3. $`\braket{cx|y} = \overline{c}\braket{x|y}`$ and $`\braket{x|cy} = \braket{x|y}c`$
  4. $`\braket{x|y} = \overline{\braket{y|x}}`$
- the inner product $`\braket{x|y}`$ is antilinear in $`x`$ and linear in $`y`$
  - if starting with two [[ket]]s, take the conjugate transpose of the former: $`\bra{x} = \ket{x}^\dagger = \overline{\ket{x}}^\intercal = \overline{\ket{x}^\intercal}`$