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 pre-hilbert_space [2013/08/31 21:04]ben pre-hilbert_space [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/08/31 21:24 nikolaj 2013/08/31 21:04 ben 2013/08/31 20:56 nikolaj 2013/08/31 20:45 nikolaj 2013/08/31 20:42 nikolaj 2013/08/31 20:42 nikolaj created2013/08/31 20:26 nikolaj 2013/08/31 20:26 nikolaj created Next revision Previous revision 2013/08/31 21:24 nikolaj 2013/08/31 21:04 ben 2013/08/31 20:56 nikolaj 2013/08/31 20:45 nikolaj 2013/08/31 20:42 nikolaj 2013/08/31 20:42 nikolaj created2013/08/31 20:26 nikolaj 2013/08/31 20:26 nikolaj created Line 1: Line 1: ===== Pre-Hilbert space ===== ===== Pre-Hilbert space ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $V$ | + | @#55CCEE: context ​    | @#55CCEE: $V$ | - | @#FFBB00: $\langle\mathcal V,​\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ | + | @#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 ==== - ==== Reference ​==== + === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Inner_product_space|Inner product space]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Inner_product_space|Inner product space]] - ==== Context ​==== + ==== Parents ​==== === Refinement of === === Refinement of === [[Vector space]] [[Vector space]] - === Requirements ​=== + === Context ​=== [[Complex number]] [[Complex number]]