Pre-Hilbert space

Set

context $V$
definiendum $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$
context $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$
context $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$
$u,v,w\in V$
$a,b\in \mathbb C$
postulate $\overline{\langle v|w \rangle}=\langle w|v \rangle$
postulate $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $
postulate $v = 0 \Rightarrow \langle v|v \rangle = 0 $
postulate $\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $
postulate $\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

Wikipedia: Inner product space

Parents

Refinement of

Context

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