Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
pre-hilbert_space [2013/08/31 20:26] nikolaj |
pre-hilbert_space [2014/03/21 11:11] (current) |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | ===== Vector space ===== | + | ===== Pre-Hilbert space ===== |
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $V$ | | + | | @#55CCEE: context | @#55CCEE: $V$ | |
| - | | @#FFBB00: $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{Pre-Hilbert}(V)$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ | |
| - | | @#88DDEE: $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ | |
| - | | @#88DDEE: $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ | |
| | $u,v,w\in V$ | | | $u,v,w\in V$ | | ||
| | $a,b\in \mathbb C$ | | | $a,b\in \mathbb C$ | | ||
| - | | @#55EE55: $\overline{\langle v|w \rangle}=\langle w|v \rangle$ | | + | | @#55EE55: postulate | @#55EE55: $\overline{\langle v|w \rangle}=\langle w|v \rangle$ | |
| - | | @#55EE55: $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ | | + | | @#55EE55: postulate | @#55EE55: $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ | |
| - | | @#55EE55: $v = 0 \Rightarrow \langle v|v \rangle = 0 $ | | + | | @#55EE55: postulate | @#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: postulate | @#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 $ | | + | | @#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 ==== | ==== Discussion ==== | ||
| - | The (pre-)Hilbert space let's us define a natural norm: | + | === Reference === |
| - | + | Wikipedia: [[http://en.wikipedia.org/wiki/Inner_product_space|Inner product space]] | |
| - | $\Vert v\Vert :=\langle v|v \rangle^\frac{1}{2}$ | + | ==== Parents ==== |
| - | ==== Reference ==== | + | === Refinement of === |
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Vector_space|Vector space]] | + | [[Vector space]] |
| - | ==== Context ==== | + | === Context === |
| - | === Subset of === | + | [[Complex number]] |
| - | [[Module]] | + | |
| - | === Requirements === | + | |
| - | [[Field]] | + | |
