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Pre-Hilbert space
Definition
$\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ |
$\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ |
$\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ |
$u,v,w\in V$ |
$a,b\in \mathbb C$ |
$\overline{\langle v|w \rangle}=\langle w|v \rangle$ |
$v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ |
$v = 0 \Rightarrow \langle v|v \rangle = 0 $ |
$\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $ |
$\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
Context
Refinement of
Requirements