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pre-hilbert_space [2013/08/31 21:04] ben |
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| - | ===== Pre-Hilbert space ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $V$ | | ||
| - | | @#FFBB00: $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ | | ||
| - | |||
| - | | @#88DDEE: $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ | | ||
| - | | @#88DDEE: $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ | | ||
| - | |||
| - | | $u,v,w\in V$ | | ||
| - | | $a,b\in \mathbb C$ | | ||
| - | |||
| - | | @#55EE55: $\overline{\langle v|w \rangle}=\langle w|v \rangle$ | | ||
| - | | @#55EE55: $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ | | ||
| - | | @#55EE55: $v = 0 \Rightarrow \langle v|v \rangle = 0 $ | | ||
| - | | @#55EE55: $\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $ | | ||
| - | | @#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]] | ||
| - | ==== Context ==== | ||
| - | === Refinement of === | ||
| - | [[Vector space]] | ||
| - | === Requirements === | ||
| - | [[Complex number]] | ||
