Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
pre-hilbert_space [2013/08/31 21:24]
nikolaj
pre-hilbert_space [2014/03/21 11:11] (current)
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]]
Link to graph
Log In
Improvements of the human condition