Inner Product

• operates on two vectors and produces a scalar
• notation $\braket{x|y}$
  • $\bra{x}$ is a [[bra]]
  • $\ket{y}$ is a [[ket]]
• [[matrix-multiplication]], associates with [[outer-product]]
• 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: $\bra{x} = \ket{x}^\dagger = \overline{\ket{x}}^\intercal = \overline{\ket{x}^\intercal}$