Week of 1/21/08

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Revision as of 15:34, 21 January 2008

$\langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle$

$\langle u,v+w \rangle \equiv langle u,v \rangle + \langle u,w \rangle$

$\langle \alpha u,v \rangle \equiv \alpha \langle u,v \rangle$

$\langle u,\alpha v \rangle \equiv \alpha^* \langle u,v \rangle$

$\langle u,v \rangle \equiv \langle v,u \rangle ^*$

$\langle u,u \rangle = 0,$ with equality only if $u = = 0$

The basic example is the form

$h(z,w) \equiv \sum z_i w^* _i$

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 [http://mathworld.wolfram.com/HermitianInnerProduct.html Definition of complex (Hermitian) inner product]
- <math> \langle u+v,w \rangle \equiv <u,w>+<v,w></math>
+<math> \langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle </math>
-<math><u,v+w> \equiv <u,v>+<u,w></math>
+<math> \langle u,v+w \rangle  \equiv langle u,v \rangle + \langle u,w \rangle </math>
-<math><\alpha u,v> \equiv \alpha<u,v></math>
+<math> \langle \alpha u,v \rangle  \equiv \alpha \langle u,v \rangle </math>
-<math><u,\alpha v> \equiv \alpha^* <u,v></math>
+<math> \langle u,\alpha v \rangle \equiv \alpha^* \langle u,v \rangle </math>
-<math> <u,v> \equiv <v,u> ^* </math>
+<math> \langle u,v \rangle  \equiv \langle v,u \rangle ^* </math>
-<math> <u,u>> = 0, </math> with equality only if <math>u==0</math>
+<math> \langle u,u \rangle  = 0, </math> with equality only if <math>u==0</math>
 The basic example is the form
 <math>h(z,w) \equiv \sum z_i w^* _i</math>