===== Pre-Hilbert space =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V$ |

| @#FFBB00: definiendum | @#FFBB00: $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ |

| @#55CCEE: context     | @#55CCEE: $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ |
| @#55CCEE: context     | @#55CCEE: $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ |

| $u,v,w\in V$ |
| $a,b\in \mathbb C$ |

| @#55EE55: postulate   | @#55EE55: $\overline{\langle v|w \rangle}=\langle w|v \rangle$ |
| @#55EE55: postulate   | @#55EE55: $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ |
| @#55EE55: postulate   | @#55EE55: $v = 0 \Rightarrow \langle v|v \rangle = 0 $ |
| @#55EE55: postulate   | @#55EE55: $\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $ |
| @#55EE55: postulate   | @#55EE55: $\langle a\cdot v+b\cdot w | u \rangle = \overline a\cdot \langle v|u \rangle+\overline b \cdot \langle w|u \rangle $ |

==== Discussion ====
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Inner_product_space|Inner product space]]
==== Parents ====
=== Refinement of ===
[[Vector space]]
=== Context ===
[[Complex number]]